Coronary capacitance

Progress in

Cardiovascular VOL XxX1,

Diseases

NO 1

JULY/AUGUST

Coronary

1988

Capacitance

Robert E. Mates, Francis J. Klocke, and John M. Canty, Jr

A

LL BLOOD VESSELS are to some degree elastic and thus expand and contract as pressure changes. As arterial pressure increases during systole, the aorta and other large arteries in the circulation expand and store blood. During diastole this stored blood is released, maintaining arterial pressure and providing a continuous flow of blood. The arterial capacitance acts as a filter, damping out the pressure and flow oscillations caused by the heart. The importance of arterial capacitance in systemic pressure-flow relationships was recognized early in the study of physiology’,2 and was first expressed mathematically in 1899 in Frank’s famous Windkessel model.’ Milnor4 has reviewed the early studies of systemic capacitance. Since the coronary circulation uses only a small fraction of cardiac output, coronary arterial capacitance is not essential in maintaining diastolic coronary flow. Phasic aortic pressure is relatively independent of coronary flow. When a major coronary arterial branch is clamped, distal pressure decreases rapidly, indicating that coronary arterial capacitance alone is not sufficient to provide much diastolic flow. Probably because of this fact, the coronary circulation until recently has been generally regarded as a purely resistive system, with coronary resistance defined as the instantaneous ratio of Brterial pressure to coronary inflow. Bellamy’s surprising finding that coronary inflow in prolonged diastoles ceased at relatively high levels of inflow pressures has led to an extensive reexamination of coronary pressureflow relationships. The dynamic nature of these relationships is illustrated in Fig 1. Coronary artery inflow is plotted as a function of arterial pressure during a prolonged diastole in which the rate of change of arterial pressure was controlled. Progress

in Czwdiovascular

Diseases, Vol XXXI,

No 1, (July/August),

The flow at any inlet pressure is seen to depend strongly on the rate of pressure change, a phenomenon that cannot be explained on the basis of a purely resistive system. Thus, although coronary capacitance does not play a major role in maintaining diastolic flow, capacitance must be accounted for in interpreting coronary pressureflow relationships. This article begins with the definition of capacitance, a review of the mechanics of distensible vessels, and a discussion of the various parameters that have been used to describe vessel compliance. The relationship between individual vessel properties and the capacitance of the coronary bed is then discussed. Measurements of coronary capacitance using a variety of techniques are summarized. Finally, the effect of capacitance on coronary pressure-flow relationships and the implications of capacitance in the clinical interpretation of coronary pressure-flow relationships are discussed. DEFINITION

OF CAPACITANCE

In a network of rigid tubes, flow at any instant of time must be the same at each cross section in the network. If the tubes are elastic and the Bow is pulsatile, periodic expansion and contraction of individual tubes alters the instantaneous flows at From the Departments of Mechanical and Aerospace Engineering, Medicine and Physiology. State University of New York at Buffalo. Supported by grants from the National Heart, Lung and Blood Institute (2-POI-HLB-15194, I-KO8-HLB-01168) and the American Heart Association (83-717). Address reprint requests to Robert E. Mates. PhD, CCI72 SUNYAB Clinical Center, 462 Grider St, Buffalo. NY 14215. 0 1988 by Grune & Stratton, Inc. 0033-0620/3006-0001$5.00/0 1988:

pp 1-15

MATES,

400-

CONSTANT -

DECLINING

360360-

CORONARY ARTERY INFLOW

240 240-

PRESSURE

PERFUSION

KLOCKE,

AND

CANT-f

= -O-

PRESSURE PERFUSION: dP/dt (mm Hg/sec) -35 =._.--TO=-125 = ..__

1

(ml/min)

CORONARY ARTERY PRESSURE (mm Hg) Fig 1. Influence of reactive elements of impedance on diastolic pressure-inflow relationships in a canine circumflex bed vasodilatad with adenosine. The thin lines represent instantaneous relationships recorded at 20-ms intervals during a single long dlastole in which coronary inflow pressure was oontrolled by a programmable source and made to decrease linearly at rates of -36, -70, and -126 mm Hg/s. The open circles represent steady state flows during eight additional long diistoles in which coronary inflow pressure was held constant at each of eight preselected levels. The curvilinear dashed line connecting the eight points is teken to be free of reactive effects and to represent the purely resistive pressure-flow relationship. The -36 mm Hg/s data are better described by a secondthan first-order fit (P < .Ol 1. whereas the -70 and - 125 mm Hg/s data sets are linear. (Reprinted with permission.‘)

different points in the system. While the mean or time averaged flows must be equal, instantaneous flow patterns at different points in the network may be quite different. This is illustrated schematically in Fig 2, which depicts a classic Windkessel in series with a rigid capillary tube. The capacitance, or compliance (C), of the Windkessel is represented by C = dV/dP, where V is volume and P is the pressure in the Windkessel. The resistance of the large tube is assumed negligible so that P is also the pressure felt by the capillary tube. At any instant, the inflow Q is divided into a capacitive flow Qc and a resistive flow QR.

flow will be positive and inflow will exceed flow through the capillary. When pressure is declining, capillary flow will exceed inflow. The magnitude of capacitive flow at any instant depends on the size of the capacitance and the rate at which pressure is changing. If the flow is periodic, the net capacitive flow over one cycle will be zero, since there is no gain or loss of flow due to the capacitor, but only temporary storage. If P,, as well as P varies with time, the instantaneous relationship between the flows will be further altered. Electrical analogs are frequently used to describe hydraulic systems. If voltage is taken as A.

Q = Qc + QR The capacitive flow is given by Qc = dV/dt = dV/dP

. dP/dt = C . dP/dt

where t represents time. If flow through capillary tube is given by

the

QR= U’ - 6,)/R where R is the tube resistance and Pb is the back pressure at the tube exit, the equation describing the relationship between inflow Q and pressure P is Q = C . dP/dt

+ (P - P,)/R

When the pressure P is increasing, capacitive

Pb QR

B.

I

I

Fig 2. (AI The hydraulic analog of a classic model of an elastic vessel. (6) The corresponding analog. See text for nomenclature and details.

Windkessel electrical

CORONARY

CAPACITANCE

3

the analog of pressure and current as the analog of flow, the electrical analog of the model is a resistor in parallel with a capacitor. This analog is also shown in Fig 2. In the coronary circulation, not only is the perfusion pressure pulsatile, but myocardial contraction impedes systolic inflow. Epicardial arteries are subjected to internal variations in coronary arterial pressure, while pressure outside the vessels is relatively constant during the cardiac cycle. Intramyocardial vessels, on the other hand, are surrounded by contracting myocardium and see quite a different pressure environment. Surface veins experience still other pressure variations. It is the transmural pressure difference that governs the magnitude of capacitive flow. Therefore the simple Windkessel model of Fig 2 is inadequate to describe the coronary circulation throughout the cardiac cycle. MECHANICS

OF DISTENSIBLE

VESSELS

The definition of capacitance (C = dV/dP) can be applied to a single vessel or to a network of vessels. If the vessel or network is occluded at its distal end, capacitance may be determined directly by measuring the volume increment required to produce a given change in pressure. Small increments should be used, since the pressure-volume relationship is generally nonlinear. In addition, the time rate of change of pressure or volume will also affect the result, as discussed later. In beds that are not occluded, capacitance cannot be determined directly, since the bed cannot be pressurized. Indirect techniques applicable in this situation are discussed later. The term volume elasticity has also been used to describe vessel distensibility. Some authors’j,’ use the term to describe capacitance, as defined above. Otherss99 define volume elasticity in terms of the relative change in volume dV/V: Eva, = dP/(dV/V)

= V/C

Determination of capacitance from this latter definition requires a measurement of the volume V of the portion of the vascular bed in question. Elastic Properties of Vessels

The behavior of elastic materials is described by the relationship between the force applied per unit area, or stress, and the fractional change in a given dimension, or strain. In general, the rela-

tionships are quite complex. However, for isotropic materials, which have the same properties in all directions, the stress-strain behavior is completely described by two independent material properties, the Young’s modulus and the Poissan ratio.4 The Young’s modulus, or modulus of elasticity, E, is the ratio of tensile or compressive stress to strain in the same direction. For nonlinear elastic materials such as blood vessels, E is not constant but varies with stress and strain. As pressure is increased, coronary arteries become stiffer,8*9 as do other arteries.“*” The Poissan ratio, v, is defined as the ratio of strain perpendicular to the direction of a stress to strain in the direction of the stress. If an elastic material is stretched in one direction, it will elongate in that direction and shorten in directions perpendicular to the direction of the stress. Blood vessel walls are nearly incompressible”; for such materials, v = 0.5. Blood vessels are not isotropic.9”33’4 For coronary arteries the modulus of elasticity along the axis is generally greater than that around the circumference. In contrast, the opposite behavior is found in the carotid artery.’ It is generally thought that tethering of vessels to surrounding tissue prevents significant length changes due to internal pressure variations, and anisotropic behavior is ignored in most analyses. The anisotropic elastic properties of the aorta were found to change significantly when the vessel was excised,13 which suggests that tethering may affect mechanical properties. Vessels embedded in the myocardium must undergo changes in length as the heart contracts. The effect of these changes has not been investigated in any detail. Several indices of vessel elasticity have been proposed. A comprehensive review of vascular wall properties and indices of elasticity is given by Milnor.4 A brief summary of these indices and their relation to capacitance is given below. Incremental elastic modulus Ei,. This index can be calculated from changes in vessel dimensions in response to pressure changes. For an incompressible vessel of constant length it can be shown that4: 3 RzRi

dP

J&c = 2 (e - R:) a

where R, is outside vessel radius, Ri is inside radius, and P is pressure. If R, is replaced by (Ri + h), where h is wall thickness, and it is

MATES,

4

noted that at constant length dV/V = 2 dRi/Ri, the equation can be rewritten: 3 (Ri + h)*V -dP Einc = [(Ri + h)* - Rz] dV Capacitance is then given by: 3 (Ri + h)* V [(Ri + h)’ - Rz] Einc

c2!!= dP

Pressure-strain modulus E,. This index is defined by E, = dP/(dD,/D,) where D, is outside vessel diameter and P is pressure. It can be shown that E, and Einc are related by the following equation”: 3 R: Einc = 2[(R, + h)* - Rf] Ep Therefore capacitance is defined in terms of E, by: c

=

2(Ri

+

Ri

h)*

V

E,

Linearized stifness parameter [. Hayashi et ali6 have recently shown that the pressure-external diameter relationship for excised coronary arteries can be linearized by the logarithmic relationship:

where P, and D, are the pressure and diameter in a reference state taken as 100 mm Hg. This provides a convenient single parameter 5 to characterize elastic behavior over a range of pressure. Wave speed c,. If pressure is changed at one end of a rigid tube filled with an incompressible fluid, the pressure change is instantly transmitted along the entire tube. On the other hand, if the fluid is compressible, or the tube is elastic, the pressure change propagates along the tube at a finite speed related to the elasticity of the tube. Neglecting frictional effects in the fluid and in the wall of the vessel, the wave speed c, is given by”: c: = dP/(p dA/A). In this expression, dA represents the local area change caused by the pressure change dP and p is the fluid density. For a tube of constant length dV/V = dA/A. Capacitance can then be calcu-

KLOCKE,

AND

CANTY

lated from the following relationship: c = V/b

6)

The definition of capacitance implies that a single value of pressure characterizes a vessel, or network of vessels, at any instant of time. For a finite wave speed this is not the case. The practical significance of wave propagation depends on how rapidly the pressure in a vessel is changed. If changes take place slowly, pressure waves will reflect back and forth in the vessel very rapidly until pressure is equilibrated (waves gradually die out due to viscous effects). The ratio of the wavelength (wave speed divided by frequency) to the vessel length determines the importance of wave propagation effects. If this ratio is much larger than one, a single value of pressure is appropriate. Of course, it is not possible to measure wave speed unless the pressure fluctuations are rapid enough to be detected at any point. Since wave speed is relatively insensitive to frequency, rapid fluctuations can be used to determine wave speed, and the calculated capacitance applied in situations in which the frequency is lower. Lumped parameter models, such as the one shown in Fig 2, are frequently used to describe portions of vascular beds because of their mathematical simplicity. They neglect the effects associated with wave propagation and reflection. The importance of wave phenomena depends on the ratio of wavelength to vessel length. Approximately 10 harmonics of the fundamental frequency are required to accurately describe coronary pressure and flow signals.4 The wavespeed in the canine coronary circulation is approximately 8 m/s.‘” At a heart rate of 1 beat/s, the tenth harmonic has a frequency of 10 Hz and a wavelength of 0.8 m, or 80 cm. Thus, vessels in the coronary circulation are short compared to the wavelengths of physiologic interest, and the use of lumped parameter models is reasonable for most applications. The various indices describing vessel elasticity are defined in terms of normalized vessel dimensions and hence are independent of the size of the specimen tested. They are thus dependent only on material properties. Capacitance, on the other hand, is proportional to the volume of the vessel. It is not possible to calculate capacitance from the other parameters unless the volume of the

CORONARY

5

CAPACITANCE

vessel or network of vessels is known. It is possible, however, to assess the probable influence of variables such as pressure and smooth muscle tone on capacitance by using the relationships obtained from studies of vessel material properties. Viscoelastic E$ects Like most other soft biologic tissues, blood vessels are not purely elastic but exhibit timedependent force-length characteristics.8~‘4*‘5*19 If a vessel segment is subjected to a sudden increase in pressure, its diameter increases as the pressure changes and then slowly increases at constant pressure, asymptotically approaching a final value. This phenomenon is called creep. Conversely, if vessel volume is suddenly increased, pressure increases rapidly and then decreases to a final equilibrium level, an effect described as stress relaxation. In vessels subjected to sinusoidally varying pressures, the variations in vessel diameter will depend on the frequency of the pressure oscillations.8 At higher frequencies the vessel appears stiffer. The diameter oscillations are also out of phase with the pressure oscillations, with diameter lagging pressure. An oversimplified explanation of this phenomenon is that the motion of fluid within the vessel wall, and hence wall deformation, is resisted by viscous forces. This time-dependent behavior is termed viscoelasticity and is discussed by Milnor.4 A simple mechanical model that exhibits viscoelastic behavior is shown in Fig 3. In this Voigt

A.

Q-

P

6. I

I

model, a viscous damper K has been added in parallel with the elastic spring C modifying the Windkessel model shown previously. At low frequency the behavior of the Voigt model is similar to the Windkessel. As frequency increases, the effective capacitance decreases until, at high frequencies, the damping limits capacitive flow. The electrical analog of the model is also shown in Fig 3. Gow et al* found the dynamic stiffness of canine coronary arteries to be about twice as large as the static stiffness at a frequency of 1 Hz, increasing to about 2.8 times the static value at 10 Hz. The Voigt model predicts a greater increase in stiffness with increasing frequency. Douglas and Greenfield” found a much larger ratio of dynamic to static stiffness in coronary arteries; however, their static measurements were made radiographically and may not be comparable. We have been successful in describing input impedance data in the coronary circulation using the model of Fig 3.21 CAPACITANCE OF THE CORONARY CIRCULATION

A vascular bed such as the coronary circulation is composed of a network of vessels of widely differing size and properties. If the dimensions and properties of each vessel were known, the static capacitance of the vascular bed could be calculated by summing the elasticity of all of the vessels. Even if such a calculation were technically feasible, the capacitance determined in this manner would have limited utility in hemodynamic calculations. The effect of the elasticity of a particular vessel on overall dynamic capacitance depends on its location in the circulation. Consider the arrangement of vessels shown in Fig 4. Vessels A and C are imagined to have negligible resistance but significant capacitance, while vessels B and D have high resistance but negligible capacitance. The pressure drop across resistance vessel B is proportional to flow

I

f- 1: b’ Fig 3. (A) The hydraulic analog of a Voigt viscoelastic element in series with a rigid capillary tube. (B) The corresponding electrical analog. See text for nomenclature and deteils.

n A

Fig 4. connected D. When capacitance

I B

C

D

Hydraulic analog of two elastic vessels, A and C. in series with two rigid resistance vessels. B and a pulsatile pressure is applied, the effective of C is diminished because of resistence B.

6

MATES,

through the resistance; a higher flow will produce a higher pressure drop. Thus pressure variations applied to vessel C will be attenuated in proportion to the resistance of B. If the resistance of B were zero, the dynamic capacitance of the system would equal the sum of the capacitances of vessels A and C. As the resistance of B is increased, the effective capacitance decreases until, for very high values of resistance B, it approaches the capacitance of vessel A. All of the vessels in the coronary bed are to some degree elastic. The effective capacitance of the bed depends on the distribution of pressure in the vascular bed. In the absence of significant stenosis, the large epicardial coronary arteries lie proximal to any appreciable resistance. Their elasticity is of major importance in determining the phasic pattern of coronary inflow produced by arterial pressure fluctuations. The elasticity of more distal arterial segments probably affects the inflow pattern as well, although their quantitative importance has not been investigated in detail. Vessels distal to the precapillary resistance, which effectively filters the inlet pressure perturbations, probably have little effect on coronary inflow during diastole. In addition to effects of arterial pressure fluctuations, intramyocardial coronary vessels are subjected to fluctuations in extravascular (intramyocardial) pressure due to the contracting myocardium. Thus intramyocardial arteries, capillaries, and veins experience quite different transmural pressure fluctuations than do surface arteries and veins. MEASUREMENTS OF CORONARY CAPACITANCE

Values for coronary capacitance have been obtained using a variety of techniques. Static capacitance can be determined only by occluding some portion of the bed, since otherwise it is impossible to maintain pressure in the absence of flow. Dynamic values of capacitance can be obtained in occluded beds by varying the distending pressure at different rates. Estimates of capacitance have also been obtained by measuring weight changes in isolated hearts. Recently, measurements of venous outflow have been used to estimate intramyocardial capacitance. Other reported techniques utilize perturbations in pressure or flow and depend on an assumed model to

KLOCKE,

AND

CANTY

calculate capacitance. Temporal variations in impedance during the cardiac cycle preclude the frequency domain analysis of pressure and flow waveforms from normal cardiac cycles (an analysis that has been frequently used in other beds to estimate capacitance). Values of capacitance obtained experimentally in the canine coronary circulation are summarized in Table 1. Results are grouped according to the method used in the experiments. Where specific models were employed to calculate capacitance, the assumed model is noted. Values of capacitance are separated into those studies in which the coronary bed had normal smooth muscle tone (tone present) and those in which the bed was vasodilated (tone absent). The last column indicates whether the calculated capacitance was tone dependent. Where the entry is blank, pressure dependence was not investigated. These studies are discussed further below. Occlusion of the Vascular Bed

Gregg et al6 were apparently the first to report measurements of coronary capacitance. They occluded the left anterior descending bed of dogs with a suspension of lycopodium, immediately postmortem. Vessels were filled with mercury to minimize flow and statically pressurized. The pressure-volume relationship was found to be linear over the range 20 to 100 mm Hg. At higher pressures the bed became less compliant. Douglas and Greenfield*’ occluded the left main coronary, left circumflex, left anterior descending, and right coronary arteries with a mixture of 200-pm beads and silicone. At initial distending pressures of 6 to 120 mm Hg, dynamic compliance was measured by injecting flow pulses to produce 50 to 75 mm Hg pressure increases over 150 to 240 ms. Since some inflow persisted at constant pressure, the RC model shown in Fig 1 was used to determine capacitance, R being determined during constantpressure perfusion. Their results showed capacitance to vary inversely with distending pressure. Eng and Kirk** cannulated the left anterior descending artery and occluded the bed with 25-pm spheres. With the inflow occluded, the bed was equilibrated to an initial pressure by collateral vessels. The cannula was then opened to a selected back pressure and capacitive dis-

CORONARY

CAPACITANCE

Table

Method Arterial

1.

Estimates

of Coronary

Mea” Distending PresSWe (mm H,)

MO&l

Capacitance Capacitance fmL/mm Hg/lOCl

Tone Present*

g)

Tons Absentf

Pressum-depsnrJent

Occlusion

Lycopodium In vitro’

spores

20-100

200 firn emboli In vitro”

IX

0.0038$

Above

go- 1505

0.0018-0.0011~~

Yes

<73n

0.014-0.017

No

25 pm emboli

100

mm Hg

In viva” Inflow Perturbation Inflow occlusion

RCRC

Arterial Inflow

pressure occlusion

decay”

Arterial

pressure

decay”

90

RCRC

0.07

Step change in pressure Long diastoler’ Sinusoidal pressure

RCR

oscillation Long diastole”

RC

30-l

Viscoelastic

30-l

Sinusoidal

Ramp pressure Long diastole’” Volume Heart

Pressure Pressure

perturbation

Venous

Long Long

RC

10

0.006-0.0016

0.014-0.0027

Yes

10

0.007-0.0017

0.022-0.0027

Yes

0.023-0.006#

Yes

20-80

change” change”

60-160

0.13**

30-125 53 53

occlusion30

change”

Outflow

Arterial Beating

0.025

Change-Isolated

Flow char&’ Coronary sinus Pressure

0.0085

pressure

oscillation Long diastole”

Blood

0.003

70-l

70

No

0.07TT

No

Measurement

occlusion hear?

0.1-0.25

diastoles3 diastole”

lo-31 30-90

*Values

were

obtained

with

TValues

were

obtained

while

normal

blood

the blood

vessel vessels

SCalculated from the data in Fig 1, assuming reported as 100 to 150 g. An average of 125 and coronary $lncrements

No

0.10 0.08 0.28

flow. Fed Proc 40:546, of 0.1 mL in volume

smooth were

muscle

tone,

13

0.1 E-0.22 0.08-o. 14

eg, the circulation

was autoregulating.

the left anterior descending bed to constitute 32% g was used. (Data from Scheal et al: The relationship

Table 1, we estimate pressure increments associated has therefore been taken as 30 mm Hg greater than I/Calculated from the data in Table 1, assuming 39%. respectively, of total cardiac mass.e

Yes

vasodilated.

1981 .I were superimposed

on initial

distending

with the volume initial distending the left main,

qDistending pressure averaged 73 ? 10 (SD) mm Hg prior anterior descending artery to preselected back pressures. #Data from Table 2, for a rate of pressure l *Calculated from the data in Fig 1 using TTCalculated from Fig 3.

0.07-O.

change dP/dt a heart weight

pressures

increments pressure.

anterior

descending,

to 1 l- to 90-mm

of total cardiac mass. Total heart weight between myocardial perfusion territory

of 60 to 120

to have

averaged

and Hg step

circumflex reductions

= 30 mm Hg . second-‘. of 120 g. Data are for one animal

only.

mm Hg. From the data 60 mm Hg. Mean beds produced

in Fig 2 and

distending

to constitute by opening

83%.

pressure 32%.

the embolized

and

8

charge calculated from the difference between retrograde flows collected in the first 30 seconds and in subsequent 30-second periods. This volume difference divided by the pressure change yields a value for capacitance. Inital distending pressures averaged 73 mm Hg and pressure steps of 11 to 90 mm Hg were employed. In contrast to the above studies, no variation of capacitance with pressure was noted. Inflow Perturbations

Lewi and Schaper23 superimposed low frequency (0.25 Hz) oscillations on aortic pressure in the beating heart. They assumed constant values for dP/dt during diastole and calculated capacitance from the variations in diastolic inflow with pressure using an RC model. Only two sets of data are presented in detail, and they show wide variability. Since heart weights were not given, it is difficult to compare their data with other reported values. Values of capacitance obtained with vasomotor tone intact were approximately three times lower than those obtained during vasodilation. The pressure dependence of capacitance was not investigated. Reported values were obtained over pressures of 50 to 150 mm Hg. Spaan et a124estimated capacitance from the time constant of the decay of arterial pressure following inflow occlusion of the left main coronary artery. They observed a rapid decay (t = 0.02 second) followed by a more gradual reduction in arterial pressure (t = 3 second). Using a model similar to that shown in Fig 4, they estimated an “epicardial” and an “intramyocardial” component of capacitance, the latter approximately 30 times as large as the former. The experiments were performed in an autoregulating bed, however, and it is likely that there was a change in vasomotor tone during the slow decay.*’ Lee et al25 used step changes in pressure during a long diastole to estimate capacitance. Resulting flow transients were analyzed using an RCR model. Capacitance was higher by a factor of two with vasomotor tone intact than in the vasodilated bed. The researchers noted a slight decrease in capacitance in the vasodilated bed when step increases rather than step decreases in pressure from the same initial level were

MATES,

KLOCKE.

AND

CANTY

employed. The study did not examine the pressure dependence of capacitance in detail. Canty et al*’ perfused the cannulated left circumflex coronary bed during prolonged diastoles with sinusoidal variations in pressure superimposed on a constant mean value. Both an RC model and a resistance in parallel with a Voigt viscoelastic element (RC viscoelastic) were used to compute capacitance. The RC viscoelastic model provided a better fit to the data; however, capacitance values obtained with both models showed similar trends. The results obtained using the RC model are summarized in Fig 5 along with the values obtained by Douglas and Greenfield” during inflow occlusion. Both sets of data show a decrease in capacitance as mean pressure was increased. The rate of change of capacitance with pressure was greatest at low pressure. In Canty’s experiment, capacitance at any given pressure in the autoregulating bed was higher by a factor of two than that obtained when the bed was vasodilated with adenosine.

l

Canty

et al

(21)

Vasodilated

0 Canty et al (21) Tone n Douglas and Greenfield Mean

OLI

,

f

Intact (20)

I SEM

1

r

30

70

CORONARY

II0

150

INFLOW PRESSURE Hg)

(mm

Fig 5. Coronary capacitance as a function of inflow pressure and vasomotor tone. Data from Canty at al*’ (their Fig 7) and Douglas and GreenMId.’ Data from Douglas and Greenfield were calculated from the data in (their) Table 1, assuming the left main, anterior desoendlng, and circurnffex beds to constitute 83%. 33%. and 39%. respectively, of total cardiac mass. Mean distending pressure was calculated assuming a pressure pulse of 80 mm Hg. (Data from Scheel et sl: The relationship between mykardlal perfusion territory and coronary flow. Fed Proe 40:545,19%1.)

CORONAAY

CAPACITANCE

Recently, Canty et al estimated capacitance in the vasodilated bed in the same preparation, utilizing down and up pressure ramps at constant dP/dt rather than sinusoidal perturbations.26 This technique allows calculation of capacitance as a function of pressure during a single long diastole. At slow ramp speeds, where viscoelastic effects are small, capacitance can be calculated from the RC model by comparing inflow during down and up ramps at the same pressure level. The calculated capacitance agreed well with that obtained using the sinusoidal perturbations. At any given pressure, apparant capacitance decreased as dP/dt increased, consistent with the viscoelastic model. Westerhof and Krams*’ have pointed out that the RC viscoelastic model employed by Canty et al” and the RCR model of Lee et al*’ have the same transfer function and hence are dynamically equivalent. Lumped parameter models such as these greatly oversimplify the complex anatomy of the coronary bed. Canty et al postulated a viscoelastic model based on the observed viscoelastic behavior of epicardial coronary arteries.’ Lee et al selected the RCR model based on their belief that most arterial capacitance lies in intramyocardial vessels. This concept was felt to be supported by an observation that diastolic transients in inflow in the beating heart perfused at constant pressure have the same time constants as those produced by step changes in pressure during long diastoles. In the former case, since arterial pressure is constant, only the intramyocardial vessels experience a transmural pressure change. In the latter case, all arterial vessels are effected. The fact that the RC viscoelastic and RCR models are equivalent emphasizes that caution must be employed in ascribing anatomic significance to individual components of lumped models. Myocardial

Blood Volume Determination

Coronary capacitance has also been estimated by measuring changes in intramyocardial blood volume. Since changes in arterial pressure will alter intravascular pressures by variable and undetermined amounts, the capacitance values obtained represent a weighted average for the entire circulation. Epicardial arterial vessels are exposed to the full change in arterial pressure;

9

more distal vessels experience smaller changes, inversely proportionai to the resistance located proximal to the vessel. Moe et al** were apparently the first to report changes in blood volume as a function of perfusion pressure. They measured weight changes in isolated beating hearts. Data from only one experiment are presented in their brief report. Diastolic heart weight was a linear function of perfusion pressure over the range 60 to 160 mm Hg. Salisbury et a129 also measured weight changes in isolated beating hearts as arterial pressure was changed. They presented average values for the entire cardiac cycle. Increases in heart rate and contractility were shown to reduce mean heart weight. This suggests that intramyocardial vessels, which are directly exposed to compressive stresses, contribute significantly to total capacitance. Salisbury et al also estimated intramyocardial blood volume after the heart was excised using red cells tagged with radioactive s1Cr.29 The value obtained represented approximately 25% of mean blood volume at 100 mm Hg in the beating heart. Thus even at very low perfusion pressures, a significant volume of blood remains in the circulation. The interpretation that weight changes are representative of intravascular volume assumes negligible changes in interstitial fluid volume. Salisbury et al showed that following inflow occlusion, weight returned to a constant baseline value within one minute. They felt that this indicated absence of interstitial volume changes. The internal chambers of the heart were also vented to prevent any accumulation of blood. In a similar isolated heart preparation, Scharf and Bromberger-Barnea3’ monitored weight changes in isolated hearts in which pressure in the cannulated left main coronary artery was increased by increasing coronary flow at constant coronary venous pressure and by coronary sinus occlusion. Since heart weights were not given, it is not possible to make an exact comparison with the data of Salisbury et al. Assuming an average left ventricular weight of 100 g, the weight changes observed when flow was increased are comparable to the values given by Salisbury. In contrast, when the coronary sinus was occluded, the weight change was four times

10

as large for comparable increases in arterial results in pressure. ‘This latter perturbation greater increases in intravascular pressure in the veins, since outflow resistance is greatly increased. The fact that the weight change was much greater with occlusion indicates that the major portion of coronary capacitance resides in distal vessels in which normal pressure is much less than arterial pressure. As discussed earlier, this experiment illustrates that the location of capacitance vessels relative to the distribution of resistance in the bed has an important bearing on their contribution to phasic pressure-flow relationships. Finally, Morgenstern et a13’ estimated blood volume changes in the beating heart from the product of coronary inflow and mean transit time measured by a dye dilution technique. Their values of capacitance, estimated as the change in blood volume with a change in coronary perfusion pressure, were similar in magnitude to those obtained using direct weight measurements outlined above. Venous Outflow Measurements Spaan’ collected great cardiac vein outflow following occlusion of the left main coronary artery. The total outflow was divided by the decrease in arterial pressure following the occlusion to estimate intramyocardial capacitance, The experiments were conducted in autoregulating beds, hence vasomotor tone was presumably decreasing during the outflow measurement. Chilian and Marcus33 measured the total coronary sinus volume flow that occurred in long diastoles during the time between cessation of arterial inflow and the end of venous outflow. Capacitance was estimated by dividing this volume by the change in arterial pressure during the same time interval when no flow was entering the system. They assumed that the coronary sinus drained the entire left ventricle. The values presented probably underestimate total capacitance. Measurements were made with normal vasomotor tone and under conditions of reduced tone following temporary inflow occlusion and asphyxia. The values obtained with reduced tone were consistently higher. Chilian and Marcus’ estimates lie between the two estimates made by Scharf and Bromberger-Barnea,30 although comparisons are tenuous, since the latter measure-

MATES,

KLDCKE,

AND

CANN

ments represent average values in the beating heart. Recently, Kajiya et a134 reported measurements of great cardiac vein flow velocity following occlusion of the left anterior descending coronary artery during adenosine vasodilation. A record from this study is shown in Fig 6. Arterial inflow was occluded for I5 seconds in the beating heart, A long diastole was then induced and the artery perfused at constant pressure. Great cardiac vein flow did not resume until after a delay of I to 2 seconds following resumption of arterial inflow. Following an initial overshoot, coronary inflow remained nearly constant during the long diastole despite the large changes in great cardiac vein flow. This result is similar to our observations in vasodilated beds3’ (Fig 3) and suggests that the effects of intramyocardial capacitance on diastolic coronary inflow are small. Kajiya et al explained their results using a model incorporating an “unstressed volume” in addition to a resistance and capacitance. During the arterial occlusion, blood is squeezed from the intramyocardial vessels, which partially or totally collapse. The refilling of these vessels requires a finite volume before any appreciable pressure develops. The unstressed volume is in effect an infinite capacitance. Anatomically, the same vessels may well be represented by the unstressed volume and the capacitance. The values of capacitance calculated by Kajiya et al are similar to those estimated from other venous outflow measuremenfs.32*33 Capacitance was found to decrease significantly when perfusion pressure was increased, while unstressed volume increased with an increase in perfusion pressure. As in the case of estimates based on weight changes, the capacitance values obtained from outflow measurements presumably represent an average, weighted by the pressure changes experienced by each vessel in the bed. At the time arterial inflow ceases, intramyocardial Row is still present and thus there will be pressure drops in the distal circulation. Vessels close to the coronary sinus will thus experience smalier pressure changes than more proximal vessels. Comparison of Capacitance Measurements Values of capacitance obtained from occlusion of the peripheral circulation and from inflow

CORONARY

11

CAPACITANCE

AoP ’ w mmlfg 3

Fig 6. Coronary inflow and outflow following inflow occlusion. AoP, aortic pressure; CPP. coronary perfusion prassure; CBF, flow in the left anterior descending coronary artery: RAP, right atrial preaaure; LVP, left ventricular pressure; GCV-V, bload velocity in the great cardiac vein. (Reorinted with oermission.g)

GC

v-v

cm/me

perturbations are, in general, more than an order of magnitude smaller than those obtained by venous outflow or blood volume measurements. The former measurements reflect primarily arterial capacitance, while the latter reflect various portions of the total bed. The much larger increase in weight observed by Scharf et a?’ when outflow was partially occluded than when a comparable increase in arterial pressure was produced by a flow increase demonstrates that the anatomic location of a capacitance has a large effect on its dynamic influence on flow. While the anatomic distribution of coronary capacitance has yet to be determined, it seems clear that only a small portion lies on the arterial side of the circulation. Decreases in vascular smooth muscle tone appear to increase measured values of capacitance regardless of the technique employed. This increase could be due to the direct effect of muscle tone on the compliance of individual vessels. It may also reflect the fact that vasodilation recruits additional vessels or exposes distal vessels to larger pressure variations. The fact that capacitance estimated from venous outflow increases with a decrease in tone33 suggests that the latter effect may be important, since vasodi-

30 *ec

1 *ec

lation is thought to occur primarily in precapillary resistance vessels. Measurements on isolated arterial segments always show an increase in stiffness as pressure is increased,**9*‘0v’6 resulting in a capacitance that decreases with increasing distending pressure. Pressure dependence of capacitance in the intact circulation is not always observed. Capacitance determined by inflow perturbation2’*26 and by occlusion with 200~pm spheres” decreased with increasing distending pressure. With occlusion of smaller vessels,6*22capacitance was independent of pressure below 100 mm Hg. In the latter studies, the entire arterial bed was subjected to a uniform pressure. Hence the elasticity of small vessels will contribute proportionally more to capacitance than when flow is present2’ and pressure drops through the circulation. The capacitance of individual coronary microvessels has not been examined. If the stiffness of these vessels is not pressure dependent, their increased contribution to the occlusion measurements might explain the discrepancy. Measurements based on blood volume changes, which include the entire circulation, do not show pressure dependence.28-30 Since the large arteries represent a small portion of total capacitance, any

12

MATES,

pressure dependence would be masked in these determinations if more distal vessels do not show pressure-dependent stiffness. Data on human coronary artery compliance is limited to postmortem measurements on excised vessels. Gow and Hadfield have shown that excised vessels are markedly stiffer than in vivo specimens. Comparisons of excised human and canine coronary vessels show similar stiffness36; however, it is not known if this similarity applies to the capacitance of the entire bed. Hayashi et alI6 suggest that the stiffness of human coronary arteries does not increase with age, in contrast to other human vessels. MODELING

CORONARY PRESSURE-FLOW RELATIONSHIPS

The relationship of observed phasic patterns of coronary inflow and outflow to flow in the microcirculation is complex and dependent on the magnitude and distribution of capacitance. While much effort has been devoted to quantifying pressure-flow relationships in recent years, considerable controversy remains.35*37 A complete description of coronary hemodynamics would require phasic pressure and flow measurements throughout the microcirculation. Optical methods have been used to measure epicardial flow patterns38s39 and a micropipet servo-null system has been applied to measure pressure in epicardial microvessels. Because of the complex distribution of ventricular stresses, pressure and flow patterns probably vary substantially from epicardium to endocardium. Extension of present experimental methods to measure transmural variations in phasic pressure and flow is a formidable task. A number of laboratories are actively developing improved techniques, which will be helpful in validating models of microcirculatory flow. The simplest model that seems adequate to describe experimental observations is shown in Fig 7. Here arterial pressure is represented by P, and venous pressure by P,. P, is the intramyocardial or extravascular pressure, and P, represents the contribution of smooth muscle tone. Arterial inflow is shown by Q., intramyocardial flow by and venous outflow by QV. Impedance is QllT7 represented by two lumped resistances (R, and R,), a Voigt viscoelastic element (C, and K), and a distal capacitance (C,). The combination of P,

KLOCKE.

AND

CANT-f

QM

Fig 7. Electrical describing phasic text for nomenclature

analog of a two coronary pressure-flow and details.

capacitance behavior.

model See

and C2 is the intramyocardial pump proposed by Spaan. Pressure distal to R, is assumed to be regulated by a waterfall-like mechanism in order to explain observed high back pressures to coronary flow.35 Unlike the original waterfall model for the coronary circulation proposed by Downey and Kirk,4’ in this model pressure is regulated only when Pb exceeds the pressure distal to the regulator. As pointed out by Spaan, this is necessary to account for observed systolic backflow when arterial pressure is reduced. The dotted lines in Fig 7 indicate control paths. We have shown*’ that smooth muscle tone P, affects the components of input impedance. Not shown on the figure is the dependence of R, and C, on arterial pressure. Panerai et a14* have shown that increases in intramyocardial pressure during systole will increase the apparent back pressure to flow. There are no doubt additional control paths that remain to be identified. During systole, increases in P, will cause capacitor C2 to discharge, increasing venous outflow and raising the pressure distal to Pb. Arterial inflow will be diminished by the increase in P,. Systolic backflow may occur if P, is sufficiently low. During systole capacitor CI will be charged as P, increases. In early diastole, as P, and Pb decrease, arterial inflow will increase as capacitor C, recharges. A discharge of C, will augment Q, during diastole. Venous flow will be diminished during diastole by the recharging of C,. While this model does not describe features of the circulation such as transmural variations in

CORONARY

13

CAPACITANCE

microcirculatory flow, it provides a conceptual framework for understanding phasic variations in arterial and venous pressure and flow throughout the cardiac cycle. As improved experimental techniques make it possible to make more detailed measurements, it will be possible to evaluate more complete models describing the impedance to coronary blood flow. It is clear that the traditional concept of coronary resistance as the ratio of arterial pressure to coronary flow is inadequate to describe phasic pressure-flow relationships. At least two separate capacitances are necessary to account for the observed phasic inflow and outflow patterns. It appears that the proximal capacitance, which affects inflow, is much smaller in magnitude than the intramyocardial capacitance. While our understanding of coronary impedance has been advanced remarkably in recent years, resolution of remaining uncertainties will require considerably more detailed measurements. CLINICAL

IMPLICATIONS

In a clinical setting, available hemodynamic data are generally limited to phasic coronary arterial pressure and inflow, which are not adequate to completely describe the pressure-flow relationships throughout the coronary criculation. Coronary resistance has frequently been employed as a measure of coronary vasodilator reserve and vascular smooth muscle tone. Many definitions of resistance have been proposed?3 some based on mean pressure and flow and some on diastolic values. From the foregoing discussion it is apparent that attempts to characterize impedance to coronary flow by a single parameter are of limited value, since the pressure-flow relationship appears to depend on several independent variables. Some of the difficulties in quantifying diastolic resistance are illustrated schematically in Fig 8. Since arterial pressure is declining during diastole, capacitive elements in the arterial circulation are discharging, and tlow in the microcirculation will exceed coronary inflow. The magnitude of this difference will depend on the rate of pressure change as well as the magnitude of the capacitance. For this reason, end- diastolic resistance (R) may differ from the true resistance of the microcirculation (R’) for three reasons: (1) back pressure to flow Pf-,, may differ from coro-

AORTIC

PRESSURE

Figure 8. Complexities of attempts to quantify diastolic resistance from measurements of arterial pressure and coronary inflow. ( 1 Diastolic coronary inflow measured during declining aortic pressure; (----I flow at the microcirculatory level. PM, end-diastolic pressure; P,. coronary venous pressure: P,, capacitance-free zero-flow pressure: I?, end-diestolic resistance derived from enddiastolic aortic pressure and coronary ffow and coronary venous pressure; R’, actual resistence et end-diistole, derived from the slope of the microcirculatory pressureflow relationship at end-diastolic aortic pressure. (Repreinted with permission.“)

nary venous pressure, (2) the slope of the pressure-flow curve in the microcirculation may differ from that of the pressure-inflow curve, and (3) the pressure-flow relationship may be nonlinear, with resistance varying inversely with pressure. Capacitive effects can be minimized by using mean full-cycle values for pressure and flo~.~ However, the resistance derived from mean values is still influenced by variations in back pressure and the curvature of the pressure-flow relationship. Further, since impedance to flow differs in systole and diastole, mean flow is also a function of heart rate!’ Mean flow will also be affected by variations in mean back pressure, caused by alterations in intramyocardial pressure or contractility. Since in patients it is not possible to obtain diastolic pressure-flow curves that are free of capacitive effects, estimates of coronary resistance must be evaluated cautiously. With presently available methods, the most accurate estimates are probably those based on the slope of the pressure-flow curve obtained form phasic aortic pressure and coronary inflow measure-

14

ments. Early diastolic values, which may reflect refilling of vessels emptied during the previous diastole, should be excluded. While values of capacitance of the human coronary circulation are not available, excised human and canine coronary arteries show similar stiffness.36 Thus it is probably reasonable to use published values for canine coronary capacitance to correct the measured inflow. In the pathologic states in which such estimates are usually made, other hemodynamic abnormalities may further complicate the interpretation. In the presence of a significant degree of coronary arterial stenosis, coronary artery pressure distal to the stenosis will differ appreciably from aortic pressure. Since phasic coronary pressure is not readily measured except during surgery, reliable estimates of coronary resistance require a knowledge of the pressure

MATES,

KLOCKE.

AND

CANTY

drop across the stenosis, which may vary significantly during diastole. In aortic stenosis, intramyocardial pressure may be abnormally elevated, thus increasing the back pressure to flow. In aortic regurgitation, the increased diastolic dp/dt will augment capacitive flow, making the correction more tenuous. The direct effect of these pathologic conditions on coronary capacitance has not been investigated. However, poststenotic dilatation or increased intramyocardial pressure may significnatly alter the capacitance of the coronary arteries. Given the assumptions involved in such estimates of coronary resistance, the significance of modest shifts in resistance must be interpreted cautiously, keeping in mind that alterations in back pressure as well as resistance are important in defining the pressure-flow relationship.

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25. Lee J, Chambers DE, Akizuki S, et al: The role of vascular capacitance in the coronary arteries. Circ Res 55:751-762, 1984 26. Canty JM Jr, Klocke FJ, Mates RE: Characterization of capacitance-free pressure-flow relationships during single diastoles in doges using an RC model with pressure dependent parameters. Circ Res 60:273-282, 1987 27. Westerhof N, Krams R: Comments on “Pressure and tone dependence of coronary diastolic input impedance and capacitance.” Am J Physiol 250:H330-H331, 1986 (letter to the editor) 28. Moe GK, Wood EH, Visscher MB: Aortic pressure and the diastolic volume law of energy output in cardiac contraction. Proc Sot Exp Biol Med 40:460-463, 1939 29. Salisbury PF, Cross CE, Rieben PA: Physiological factors influencing blood volume in isolated dog hearts. Am J Physiol200:633-636, 1961 30. Scharf SM, Bromberger-Barnea B: Influence of coronary flow and pressure on cardiac function and coronary vascular volume. Am J Physiol224:918-925, 1973 31. Morgenstern C, Holjes U, Arnold G, et al: The influence of coronary pressure and coronary flow on intracoronary blood volume and geometry of the left ventricle. PfIugers Arch 340:101-l 11, 1973 32. Spaan JAE: Intramyocardial compliance studies by venous outflow at arterial occlusion. Circulation 66:307, 198 I (suppl 3; abstr) 33. Chilian WM, Marcus ML: Coronary venous outflow persists after cessation of coronary arterial inflow. Am J Physiol247:H984-H990. 1984 34. Kajiya F, Tsujiokak, Goto M, et al: Functional characteristics of intramyocardial capacitance vesselsduring diastole in the dog. Circ Res 58:476-485, 1986 35. Klocke FJ, Mates RE, Canty JM Jr, et al: Coronary pressure-flow relationships-Controversial issues and probable implications. Circ Res 56:31&323, 1985

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36. Gow BS, Hadfield CD: The elasticity of canine and human coronary arteries with reference to post-mortem changes. Circ Res 45:588-594, 1979 37. Spaan JAE: Coronary diastolic pressure-flow relation and zero flow pressure explained on the basis of intramyocardial compliance. Circ Res 56:293-309, 1985 38. Tillmanns H, Ikeda S, Hansen H, et al: Microcirculation in the ventricle of the dog and turtle. Circ Res 34:561569,1974 39. Steinhausen M, Tillmanns H, Thederan H: Microcirculation of the epimyocardial layer of the heart. I. A method for in vivo observation of the microcirculation of superficial ventricular myocardium of the heart and capillary flow pattern under normal and hypoxic conditions. Pflugers Arch 378:9-14, 1978 40. Tillmanns H, Steinhausen M, Leinberger H, et al: Pressure measurements in the terminal vascular bed of the epimyocardium of rats and cats. Circ Res 49:1202-l 211, 1981 41. Downey JM, Kirk ES: Inhibiton of coronary blood flow by a vascular waterfall mechanism. Circ Res 36:753760,1975 42. Panerai RB, Chamberlain JH, Sayers BM: Characterization of extravascular component of coronary resistance by instantaneous pressure-flow relationships in the dog. Circ Res 45:378-390, 1979 43. Marcus M: The Coronary Circulation in Health and Disease. New York McGraw-Hill 1983 44. Vlahakes GJ, Baer RW, Uhlig PN, et al: Adrenergic influence in the coronary circulation in conscious dogs during maximum vasodilation with adenosine. Circ Res 51:371-384, 1982 45. Bathe RJ, Cobb FR: Effect of maximal coronary vasodilation on transmural myocardial perfusion during tachycardia in the awake dog. Circ Res 41:648-653, 1977