Simple model of circulatory system dynamics including heart valve mechanics T. Cochrane Department of Medical Sheffield SlO 2JF, UK
Physics
and Clinical
Received September 1990, accepted January
Engineering,
Royal Hallamshire
Hospital,
1991
ABSTRACT Thispaper describes an extension of the Windkessel model of circulatory system dynamics, which takes into account the opening action of the mitral and aortic valves, including stenotic and regurgitant orifice. The starting point for the model is the ventricular emptying/jlling curve which is taken from a quasi-physiological ventricular flow relationship which incorporates variation of systolic and diastolic intervals with qcle rate. The valves are assumed to open with a linear rise in area up to maximum orifice, followed by a period at maximum orifice and then a linearfall-off in area to the closedposition (which may allow regurgitation). Flow through the valves is assumed to be governed by the Gorlin equation. Peripheral resistance and compliance are considered asfixed parameters of the arterial system. 7he model is use@1 in l&ping to understand the complex interaction between valvular mechanics and the rest of the cardiovascular system. Applications of the model are illustrated by considering isolated aortic stenosis, isolated aortic regurgitation, cardiat adaptation in the presence of these two abnormalities and the efects of variation of perspherat resistance on pressures within the cardiovascular system. Keywords: Windkessel pressure,
ventricular
model,
pressure,
cardiovascular
cardiac
model,
stenosis,
regurgitation,
valve
orifice,
arterial
work
INTRODUCTION Despite its simplifying assumptions, the Windkessel greatly to our theory of Frank’ has contributed understanding of the dynamics of the circulation. In the last thirty years, man-made valves have increasingly been used to replace defective native valves. Indeed, the treatment of valvular heart disease now constitutes a substantial proportion of the workload of the cardiologist. A defective valve, or the introduction of a man-made alternative, represents a significant intervention within the cardiovascular system. The effects of this intervention are not easy to predict. Furthermore, the interaction of valves with different o ening actions does not seem to have been considere cf previously. This paper proposes a simple model of circulatory dynamics based on the Windkessel approach, but which takes account of valvular opening action, including stenotic and regurgitant properties. The model is first described in detail. Illustrations of the application of the model are given in subsequent sections.
container with run-off through a peripheral resistance. Inflow from the ventricle increases the pressure in the compliant arterial system, which then drives the outflow through the peri heral resistance. This establishes a new pressure an B volume in the arterial system and the process is then repeated until sufhcient iterations have been performed to reach the ulsatile situation. In the simulations steady-state discussed be Pow, 20 cardiac cycles were enough to reach this steady state. The major components of the model are illustrated in Figure 7. The model assumes that the compliance of the arterial system is constant and that peripheral flow is driven by the pressure through a constant resistance. It further assumes that there is no pressure pulse propagation or wave reflection. All of these oss simplifications of the real assumptions are situation. Nonethe ffess, prior successes with this type Compliance
\
Atria1 head
THE MODEL The starting point for the model is the simple approach used by Randall2 in a teaching simulation of the arterial pressure pulse. The arterial system, into which the heart pumps, is viewed as an elastic
esistonce
/
Aortic
Valve
Ventricle
Figure 1 Components of the Windkessel model of the circulatory system including heart valve opening behaviour
0 1991 Butterworth-Heinemann for BES 0141-542.5/91/040335-06
J. Biomed. Eng. 1091, Vol. 13, July
335
Circulatory system dynamics and heart valve mechanics: T. Cochmne Time (s) 0 0
I.0 I
, I I I I
-tm
Systale
Figure 3 Valve opening. AO,,,, maximum aortic orifice; Areg, regurgitant aortic area; t,, aortic valve opening and closing time; MO,,,, maximum mitral orifice; Mregr regurgitant mitral area; t,, mitral valve opening and closing time
a period at maximum opening, followed by a linear fall-off in area to closure or to a fixed regurgitant area. Both valves behave similarly, though with different phase.
600 r -
Diastale
Calculation
_
Time(s) I 0
I.0
b
of modelling indicate that, though the detail is missing, major trends and the order of magnitude of effects are predicted. The purpose of the model proposed here is not to reproduce exactly the situation in uivo but to help us to understand the behaviour of the cardiovascular system in the presence of non-perfect heart valves. Flow: ventricular
emptying/filling
curve
The controlling flow is taken from the ventricular emp ing/filling curve of Figure 2a, which is based on the i El? ealized ventricular flow shown in Figure 2b. The example shown represents a stroke volume of 80 ml at a heart rate of 70 beatsmin-‘. Different heart-rate/ stroke-volume combinations are obtained by appropriate scaling of systolic and diastolic time intervals (to adjust the heart rate), followed by scaling of the volume flow to give the required stroke volume. The seven element piecewise-linear model of ventricular flow shown in Figure 26 allows new heart-rateistrokevolume combinations to be computed easily. The systolic/diastolic intervals are taken from a quasiphysiological relationship which has been described previously”. Valve
opening
action
The opening actions of the aortic and mitral valves are given in Figure 3. These assume a linear rise in area during opening to maximum orifice followed by
336
J. Biomed. Eng. 1991, Vol. 13, July
and flows
The controlled parameters in all simulations is ventricular flow (Figure 2b). The volume corresponding to the first time increment is injected into the arterial system shown in Figure 1. This generates a pressure, P, within the compliant arterial system, given by: P=
Figure 2 a, Idealized ventricular emptying/filling curve; b, corresponding ventricular flow curve. SV, stroke volume ejected; DI, diastole inflow (these two volumes are equal)
of pressures
v/c
(1)
where C is the compliance of the arterial system and V is the excess volume. The run-off during this period is given by: Run-off = Pdt/R
(2)
where dt is the time increment and R is the peripheral resistance. The difference between the volume injected and the run-off is the excess volume for the given time step and this is added to the excess volume in the arterial system to generate the aortic pressure (P) for the next iteration. This process is repeated until enough iterations have been performed to reach the desired end point. The ventricular pressure which drives the flow through the valve orifices is calculated by assuming that the flow is governed by the Gorlin formula: EOA = Q/(44.5 &“‘)
(3)
where EOA is effective orifice area, Qis the ventricular flow and Pd is the pressure drop across the valve. The fact that the two valves in this simulation are not necessarily ideal complicates the further anal sis. The net ventricular flow is the sum of the flyows through the aortic and mitral valves. Consider the situation in Figure 1 where the piston is moving in the forward direction (from left to right). Most of the flow will pass through the aortic valve but some may flow through the mitral valve during closure and by regurgitant leakage. The situation is governed by the equation: uf( i) = 44.5 [(up - hd) “’ X ma(i)
+
(up - P) 1’2 X au(i)] where vf is the ventricular
(4)
flow at the ith time step, up
Circulatory q&m
dynamics and heart valve mechanics: T. Cochrane
/
------l-
--I3
HR = 40
Diostole
Rest
Figure 4
a .
Flow situations
which
are possible
in the present
orifice
( cm2)
orifice
(cm*
model
is the ventricular pressure during the time interval under consideration, hd is the atria1 pressure head, ma(i) is the mitral valve area at the ith step, P is the pressure in the arterial system (aortic pressure) and aa( i) is the aortic valve area at the ith time step. Because all of the parameters except up are known at each time step, solution of this quadratic equation allows the ventricular pressure to be calculated. There are four possible situations which need to be treated for the ‘cardiac’ cycle depicted in Figure 2b. These are illustrated in Figure 4. Equations similar to equation (4) can be written and solved for each of these situations. Complete solution of the problem allows ventricular and aortic pressures, aortic and mitral flows and associated parameters such as systolic/diastolic pressure ratio and regurgitant volumes to be calculated. In addition, the effects of valvular stenosis and regurgita tion, peripheral resistance and compliance at different heart-rate/stroke-volume combinations may be examined simply by changing one or two input parameters. Some exam les of the applicability of the model are given in the Pollowing section. APPLICATIONS
Aortic 600r
I .5
IO Aortic 000
C
1
r
0
05
1.0 Aortic
I 5 orifice
2.0
(cm*)
Figure 5 Ventricular pressure versus aortic valve maximum orifice area for stroke volumes (SY) of 40 and 100 ml at heart rates (HR) a, 40; b, 80; c, 120 beats min-‘. -, sv= 40; o-0, sv100
OF THE MODEL
Isolated aortic valve stenosis Isolated aortic valve stenosis is an increasing problem in an ageing population. The model was used to examine the effects of increasing stenosis grade on ventricular ressure for different heart rate (40120 beats m- P) and stroke volume (40- 120 ml) combinations. Because ventricular flow was controlled for each heart-rate/stroke-volume combination, for each grade of stenosis, systolic and diastolic arterial pressures remained almost unaltered and were not considered further in this example. The major effects of stenosis were seen in the ventricular pressure. Maximum mitral valve opening area was fixed at 2.0cm2 and mitral regurgitant area was 0.01 cm’. Aortic regurgitant area was also fixed at 0.01 cm’. Peripheral resistance and eripheral compliance were set at their ‘standard’ va Pues of 1.08 mmHg ml-’ s-r and 1.5 ml mmHg-’ respectively’. Both valves o ened to maximum orifice over a period of 50ms. & e time step used for each iteration, as in all the examples discussed below, was 5ms.
Figure 5 shows some representative examples of ventricular pressure plotted against stenotic valve orifice area in the range 0.5-2.0cm2. Results for heart rates of 40 beatsmin-‘, 80 beatsmin-’ and 120 beatsmin-’ are shown for stroke volumes of 40 and 100ml. Isolated aortic regurgitation This finding, too, is not uncommon with both native valves and bioprosthetic pericardial valves. The above set of heart-rate/stroke-volume simulations was repeated, only this time maximum aortic area was fixed at 2.0cm2 and the regurgitant area was varied over the range 0.01-o. 15 cm2. Figure 6 shows the effect of increasing regurgitant area on the three pressures: mean ventricular pressure over s stole, systolic arterial pressure and diastolic arteria r ressure. Figure 7 shows how the regurgitant fraction Ptotal regurgitant-volume/stroke-volume, expressed as a percentage) varies with regurgitant aortic area for
J. Biomed.
Eng. 1991, Vol. 13, July
337
Circulatory sys&m dynamics and hart ualve mechanic: 12Or
HR = 60
I
0
3.0
SV = 80
I
I
0.1
0.15
I
0.05
T. Cochrane
0
Regurgitant area (cm2 ) arterial 6 Mean ventricular pressure over systole (-), systolic pressure (0 -Cl) and arterial diastolic pressure (A-A) plotted against aortic valve regurgitant area for a heart rate (HR) of 60 beats min-’ and stroke volume (SY) of 80 ml
Figure
100
80 z
SV = 80 ml
F
r
I
I
0.05
0. I
I 0.15
Regurgitantoreo (cm2
)
Figure 8 Factor by which the work performed by the heart to deliver a givenvolume to the peripheral circulation must be increased in the ), 40ml at 40beatsmiu’; presence of aortic regurgitation. ((A-A), 60ml at 80beatsmin’; (O-Cl), 80ml at 120beatsmiu
rate/stroke-volume calculated from:
combinations.
The vertical axis is
Excess work factor = (SV& x Pj&../(sVo
x I$~o)
where SV,, and Vpregare the required stroke volume pressure in the and mean systolic ventricular presence of regurgitation and SF’,, and VP, are the stroke volume and mean systolic ventricular pressure when there is no regurgitation.
Limiting of cardiac output in isolated aortic stenosis
L
0
I
I
I
0.05
0.
I
Regurgitant urao (cm2
0.15
1
Figure 7 Regurgitant fraction plotted against aortic valve regurgitant ), 80 (A-A) and 120 beatsmin-’ area at heart rates (HR) of 40 ((O-Cl) for a stroke volume of 80 ml
heart rates 40, 80 and volume of 80ml.
Cardiac adaptation regurgitation
120 beatsmin-’
in the presence
at a stroke
J. Biomed. Eng. 1991, Vol. 13, July
120 -
of isolated
Most forms of valvular disease develop over a prolonged period and there is usually concomitant cardiac adaptation. In the simulation above, the stroke volume remained fixed and the effect of isolated regurgitation on ventricular and arterial ressures was investigated. In the real situation, the g ear-t would ada t, insofar as it was able, to deliver the net volume Kow re uired by metabolic demands. The model was adapte 8 to simulate this situation for different heart rates and net stroke delivered to the peripheral circulation (s stohc stroke-regurgitant volume). The effect of the CT ifferent levels of regurgitation is probably best demonstrated by deriving the factor by which the work done by the heart must be increased to deliver the same volume as would be delivered in the absence of regurgitation. The results are illustrated in Figure 8 for three different heart-
338
Just as the heart adapts in the presence of aortic regurgitation, it adapts to aortic stenosis. It would be interesting to know the maximum cardiac out ut that can be attained in the presence of a given Peve1 of stenosis. An approximate answer to this uestion may be obtained by assuming an upper ‘fimit to the maximum ventricular pressure that can be sustained. Having set this pressure ceiling, the maximum stroke volume which can be delivered for a given stenotic
100 _ 5 80 E j_:
/:--:::_::
; .$
40 -
P
20 I
0
0.5
I
I
I .o
1.5
Aortic orifice
1
2.0
(cm’ )
Figure 9 Maximum attainable stroke volume (for mean ventricular pressure of 200mmHg) for different grades of stenotic aortic orifice. ), 120 beats min-‘; (A - A), 80 beatsmin-’ (-
Circulatory qtem
80ml and 120 beats mini.
was
dynamics and heart valve mechanics: T. Cochrane
the heart
rates
were
40,
80
and
DISCUSSION The major
05
0.7
0.9 Peripheral
I.1
resistance
(mmHg
1.3 ml-‘s-’
I.5
)
Figure 10 Plots illustrating the linear relationship between mean ventricular pressure (), arterial systolic pressure (A -A) and arterial diastolic pressure (O--O) and peripheral resistance. Results shown are for a stroke volume of 100 ml at 60 beats min-’ without any aortic regurgitation
area, for a given heart rate, can then be calculated. In the examples illustrated in Figure 9, the maximum sustainable mean ventricular pressure was set at 200mmHg. This is an arbitrary figure, but a reasonable one for the purpose of illustration. Maximum stroke volume is plotted against re gitant area for two different heart rates, 80 Yreatsmin-’ and 120 beats min-‘.
Variation of peripheralresistance Problems of the circulation are not always related solely to the heart and heart valves. There are often associated changes to the systemic arteries. S stemic variations may be modelled by varying ei x er the peripheral resistance R, or the compliance C, or both. Figure 70 shows the variation of mean systolic ventricular, systolic arterial, and diastolic arterial pressures with peripheral resistance for a stroke volume of 100 ml at 60 beats mint. Peripheral resistance ranges from 0.7 mmHgmll’ s-‘, which is ‘hypotensive’, to 1.5 mmHg ml-’ s-l, which is ‘hypertensive’. Figure 17 demonstrates the effect of peripheral resistance on the regurgitant volume in the presence of different levels of regurgitation. The stroke volume in these examples
40
r
05
0.7
0.9 Peripheral
resistance
I .I ( mmHg
I .3 ml-‘s-l
1.5
)
Figure 11 Regurgitant volume plotted against peripheral resistance for two values of regurgitant aortic area (& Stroke volume in all 80 (A - A) and 120 cases was 80m1, heart rates were 40 (O-Cl) beatsmin~’
effect of isolated aortic stenoses is to increase the ventricular pressure during systole, Figure 5. The increase in ventricular pressure becomes dramatic when the orifice area falls below 1.0 cm2. Even at a heart rate of only 40 beats min-‘, the mean ventricular pressure required to maintain a stroke volume of lOOm1 when the aortic orifice is 0.5 cm2 rises to over 440mmHg. Such a ressure is extremely unlikely to be achieved in vivo. ?1%us, stenosis imposes a limit on the volume that can be delivered by the heart with each stroke. This is illustrated by the results of Figure 9. A maximum sustainable mean systolic ventricular pressure of 200 mmHg has been assumed in generating the two curves, for heart rates of 80 and 120 beatsmin-‘. This figure illustrates clearly the limiting effects of severe aortic stenosis on the cardiac output. For example, at 80 beatsmin-’ the maximum stroke volume that can be delivered for a stenotic orifice area of 1 cm2 is 75 ml, which gives a maximum cardiac output of 6.0litremin-‘, enough to maintain an individual at rest but not allowing much scope for exercise. The effects of acute isolated aortic regurgitation, such as might occur with leaflet tear, are examined in Figures 6 and 7. Ventricular and aortic systolic and diastolic pressures fall off almost linearly with the re rigtant area. There are differences in detail with di $ferent heart rate and stroke volume combinations, but the overall effect is the same, an almost linear falloff in all three pressures. Figure 7 is interesting because it combines the effects of different heart rates and regurgitant areas on the regurgitant fraction (percentage of stroke volume which flows back through the regurgitant valve during diastole). At 40 and 80bpm, the regurgitant fraction is dominated by leakage during the diastolic interval, the former bein worse because of the prolonged diastolic interva 4i. At 120beatsmin’, the pressure in the arterial s stem rises more sharply than at the lower rates. TZ us, although the diastolic interval as a proportion of cycle length is shorter at 120 beatsmin-* than that at 80 beatsmin-*, the increase in pressure means that the regurgitant fraction is actually higher. The volume swept back during valve closure is also increased at higher pressures. This explains why the curves for 120 and 80 beats min-’ in Figure 7 do not project back to zero for zero regurgitant area. In most cases, the heart will adapt to the regurgitant situation to deliver the necessary volume flow. To do so, it must work harder. The factor b which the workload on the ventricle is increased ?Jy regurgitation is illustrated in Figure 8 for three different heartrate/stroke-volume combinations. Not unexpectedly, the situation is worst at the lowest heart rates. For example, at 40 beatsmin-’ with a regur ‘tant area of only 0.15 cm2 the ventricle must wor !Y 2.66 times harder per beat to deliver a stroke volume of 40ml. The excess work factor is improved by increasing heart rate because the diastolic interval available for regurgitation is shortened.
J. Biomed. Eng. 1991, Vol. 13, July
339
Circulatory system dynamics and heart valve mechanics: T. Cochrane
However, expressing the results in this way belies the true situation, particularly as the stroke volume increases. As heart rate increases for a given stroke volume the pressure in the arterial system inevitably rises and the ventricle has to work harder to overcome this higher pressure. Thus, the baseline work, in the absence of regurgitation, will be higher and although the factor by which the work must be increased to cope with regurgitation is less, the ventricle may not be able to reach the workloads demanded. This is illustrated for the situation represented by the point in Figure 8 corresponding to a stroke volume of 80 ml at 120 beats min-’ with a regurgitant area of 0.15 cm’. The increased workload is only a factor of 1.39 higher et the mean ventricular ressure required is 236mm ?I g. Volume delivery is Pikel to be compromised in this situation. These resu rts assume a maximum aortic orifice of 2.0 cm’. The situation would be made worse by coexisting aortic stenosis. Figure 70 illustrates the almost linear relationship between the pressures within the system and the peripheral resistance. Under the severe hypotension represented by a peripheral resistance of the systolic pressure falls to 0.7mmHgmll’s-l, 97mmHg and the diastolic pressure drops to only 47 mmHg, pressures which would be barely adequate to support the circulation. In h ertension, produced to increasing periphera Yp resistance bY 1.5mmHgmlr s-r, systolic and diastolic pressures and the heart rise to 175 and 125 mmHg, respective1 must work harder pro ortionately to (Yeliver the flow required by the perip Kera1 tissues. In the presence of a regurgitant area, the effect of increasing peripheral resistance is to increase the regurgitant volume, Figure 17. There are two interesting points to note about this figure. As was observed when considering the acute effects of regurgitation in Figure 7, regurgitation has its worst effects when the heart rate is low and the diastolic interval, correspondingly, is long. At higher heart rates, the pressure in the arterial system rises. Thus, in certain circumstances, regurgitation is higher at a rate of 120 beats min-’ than at 80 beats min-’ though the diastolic interval, as a proportion of the cycle length, is less. However, this does not hold in all situations. For example, as the area available for regurgitation increases in tandem with an increase in peripheral
340
J. Biomed. Eng. 1991, Vol. 13, July
resistance, the 120 beats min-’ and the 80 beatsmin-’ plots cross over and the regurgitant volume at 80 beatsmin-’ becomes greater than that at This may be explained by the 120 beats min-‘. sh er fall-off in pressure during diastole as the area avai “p able for regurgitation increases. There seems to be no similar work with which to compare the model discussed here. The simplifying assumptions outlined in the description of the model limit its applicability as a simulation of the situation in vim. Its greatest value is in helping to understand the complex interaction between heart valves and the rest of the circulation. Valve opening action is modelled simplistically and variation of systolic and diastolic intervals with heart rate is not taken from an ‘exact’ physiological relationship, though it is doubtful that such a thing exists. On the other hand, the approximations made are not so far removed from reality as to be unreasonable. In fact, in our heart valve test laboratory, features predicted by the model have been observed subsequently in tests on prosthetic valves in vitro. The major limitation of the model is that it ignores the pressure wave and wave reflections and is, therefore, unable to predict the exact shape of the ressure profiles. Conversely, the prediction of flow Kehaviour, including closing and leakage regurgitation through defective valves, is a reasonable approximation of that observed in our laboratory studies of prosthetic heart valves. Features such as variation of systemic compliance or resistance with distending pressure would be relatively easy to build into the model. Further realism could be added by measuring valve opening areas directly frame-by-frame from high speed tine or video images (in vitro situation only) and thereby developing a better model of valve opening action.
REFERENCES Frank 0. Schltzung des Schlagvolumens des menschlichen Herzen auf Grund der Wellen und Windkessels theorie. 2 Biol 1930; 90: 405-9. Randall JE. Microcomputers and physiological simulation, New York: Addison-Wesley, 1980; chapter 9: 116-28. Shortland AP, Cochrane T. Doppler spectral waveform generation in vitro: an aid to diagnosis of vascular disease. Ultrasound Med G? Biol 1989;
15: 737-48.
Physics
and Clinical
Received September 1990, accepted January
Engineering,
Royal Hallamshire
Hospital,
1991
ABSTRACT Thispaper describes an extension of the Windkessel model of circulatory system dynamics, which takes into account the opening action of the mitral and aortic valves, including stenotic and regurgitant orifice. The starting point for the model is the ventricular emptying/jlling curve which is taken from a quasi-physiological ventricular flow relationship which incorporates variation of systolic and diastolic intervals with qcle rate. The valves are assumed to open with a linear rise in area up to maximum orifice, followed by a period at maximum orifice and then a linearfall-off in area to the closedposition (which may allow regurgitation). Flow through the valves is assumed to be governed by the Gorlin equation. Peripheral resistance and compliance are considered asfixed parameters of the arterial system. 7he model is use@1 in l&ping to understand the complex interaction between valvular mechanics and the rest of the cardiovascular system. Applications of the model are illustrated by considering isolated aortic stenosis, isolated aortic regurgitation, cardiat adaptation in the presence of these two abnormalities and the efects of variation of perspherat resistance on pressures within the cardiovascular system. Keywords: Windkessel pressure,
ventricular
model,
pressure,
cardiovascular
cardiac
model,
stenosis,
regurgitation,
valve
orifice,
arterial
work
INTRODUCTION Despite its simplifying assumptions, the Windkessel greatly to our theory of Frank’ has contributed understanding of the dynamics of the circulation. In the last thirty years, man-made valves have increasingly been used to replace defective native valves. Indeed, the treatment of valvular heart disease now constitutes a substantial proportion of the workload of the cardiologist. A defective valve, or the introduction of a man-made alternative, represents a significant intervention within the cardiovascular system. The effects of this intervention are not easy to predict. Furthermore, the interaction of valves with different o ening actions does not seem to have been considere cf previously. This paper proposes a simple model of circulatory dynamics based on the Windkessel approach, but which takes account of valvular opening action, including stenotic and regurgitant properties. The model is first described in detail. Illustrations of the application of the model are given in subsequent sections.
container with run-off through a peripheral resistance. Inflow from the ventricle increases the pressure in the compliant arterial system, which then drives the outflow through the peri heral resistance. This establishes a new pressure an B volume in the arterial system and the process is then repeated until sufhcient iterations have been performed to reach the ulsatile situation. In the simulations steady-state discussed be Pow, 20 cardiac cycles were enough to reach this steady state. The major components of the model are illustrated in Figure 7. The model assumes that the compliance of the arterial system is constant and that peripheral flow is driven by the pressure through a constant resistance. It further assumes that there is no pressure pulse propagation or wave reflection. All of these oss simplifications of the real assumptions are situation. Nonethe ffess, prior successes with this type Compliance
\
Atria1 head
THE MODEL The starting point for the model is the simple approach used by Randall2 in a teaching simulation of the arterial pressure pulse. The arterial system, into which the heart pumps, is viewed as an elastic
esistonce
/
Aortic
Valve
Ventricle
Figure 1 Components of the Windkessel model of the circulatory system including heart valve opening behaviour
0 1991 Butterworth-Heinemann for BES 0141-542.5/91/040335-06
J. Biomed. Eng. 1091, Vol. 13, July
335
Circulatory system dynamics and heart valve mechanics: T. Cochmne Time (s) 0 0
I.0 I
, I I I I
-tm
Systale
Figure 3 Valve opening. AO,,,, maximum aortic orifice; Areg, regurgitant aortic area; t,, aortic valve opening and closing time; MO,,,, maximum mitral orifice; Mregr regurgitant mitral area; t,, mitral valve opening and closing time
a period at maximum opening, followed by a linear fall-off in area to closure or to a fixed regurgitant area. Both valves behave similarly, though with different phase.
600 r -
Diastale
Calculation
_
Time(s) I 0
I.0
b
of modelling indicate that, though the detail is missing, major trends and the order of magnitude of effects are predicted. The purpose of the model proposed here is not to reproduce exactly the situation in uivo but to help us to understand the behaviour of the cardiovascular system in the presence of non-perfect heart valves. Flow: ventricular
emptying/filling
curve
The controlling flow is taken from the ventricular emp ing/filling curve of Figure 2a, which is based on the i El? ealized ventricular flow shown in Figure 2b. The example shown represents a stroke volume of 80 ml at a heart rate of 70 beatsmin-‘. Different heart-rate/ stroke-volume combinations are obtained by appropriate scaling of systolic and diastolic time intervals (to adjust the heart rate), followed by scaling of the volume flow to give the required stroke volume. The seven element piecewise-linear model of ventricular flow shown in Figure 26 allows new heart-rateistrokevolume combinations to be computed easily. The systolic/diastolic intervals are taken from a quasiphysiological relationship which has been described previously”. Valve
opening
action
The opening actions of the aortic and mitral valves are given in Figure 3. These assume a linear rise in area during opening to maximum orifice followed by
336
J. Biomed. Eng. 1991, Vol. 13, July
and flows
The controlled parameters in all simulations is ventricular flow (Figure 2b). The volume corresponding to the first time increment is injected into the arterial system shown in Figure 1. This generates a pressure, P, within the compliant arterial system, given by: P=
Figure 2 a, Idealized ventricular emptying/filling curve; b, corresponding ventricular flow curve. SV, stroke volume ejected; DI, diastole inflow (these two volumes are equal)
of pressures
v/c
(1)
where C is the compliance of the arterial system and V is the excess volume. The run-off during this period is given by: Run-off = Pdt/R
(2)
where dt is the time increment and R is the peripheral resistance. The difference between the volume injected and the run-off is the excess volume for the given time step and this is added to the excess volume in the arterial system to generate the aortic pressure (P) for the next iteration. This process is repeated until enough iterations have been performed to reach the desired end point. The ventricular pressure which drives the flow through the valve orifices is calculated by assuming that the flow is governed by the Gorlin formula: EOA = Q/(44.5 &“‘)
(3)
where EOA is effective orifice area, Qis the ventricular flow and Pd is the pressure drop across the valve. The fact that the two valves in this simulation are not necessarily ideal complicates the further anal sis. The net ventricular flow is the sum of the flyows through the aortic and mitral valves. Consider the situation in Figure 1 where the piston is moving in the forward direction (from left to right). Most of the flow will pass through the aortic valve but some may flow through the mitral valve during closure and by regurgitant leakage. The situation is governed by the equation: uf( i) = 44.5 [(up - hd) “’ X ma(i)
+
(up - P) 1’2 X au(i)] where vf is the ventricular
(4)
flow at the ith time step, up
Circulatory q&m
dynamics and heart valve mechanics: T. Cochrane
/
------l-
--I3
HR = 40
Diostole
Rest
Figure 4
a .
Flow situations
which
are possible
in the present
orifice
( cm2)
orifice
(cm*
model
is the ventricular pressure during the time interval under consideration, hd is the atria1 pressure head, ma(i) is the mitral valve area at the ith step, P is the pressure in the arterial system (aortic pressure) and aa( i) is the aortic valve area at the ith time step. Because all of the parameters except up are known at each time step, solution of this quadratic equation allows the ventricular pressure to be calculated. There are four possible situations which need to be treated for the ‘cardiac’ cycle depicted in Figure 2b. These are illustrated in Figure 4. Equations similar to equation (4) can be written and solved for each of these situations. Complete solution of the problem allows ventricular and aortic pressures, aortic and mitral flows and associated parameters such as systolic/diastolic pressure ratio and regurgitant volumes to be calculated. In addition, the effects of valvular stenosis and regurgita tion, peripheral resistance and compliance at different heart-rate/stroke-volume combinations may be examined simply by changing one or two input parameters. Some exam les of the applicability of the model are given in the Pollowing section. APPLICATIONS
Aortic 600r
I .5
IO Aortic 000
C
1
r
0
05
1.0 Aortic
I 5 orifice
2.0
(cm*)
Figure 5 Ventricular pressure versus aortic valve maximum orifice area for stroke volumes (SY) of 40 and 100 ml at heart rates (HR) a, 40; b, 80; c, 120 beats min-‘. -, sv= 40; o-0, sv100
OF THE MODEL
Isolated aortic valve stenosis Isolated aortic valve stenosis is an increasing problem in an ageing population. The model was used to examine the effects of increasing stenosis grade on ventricular ressure for different heart rate (40120 beats m- P) and stroke volume (40- 120 ml) combinations. Because ventricular flow was controlled for each heart-rate/stroke-volume combination, for each grade of stenosis, systolic and diastolic arterial pressures remained almost unaltered and were not considered further in this example. The major effects of stenosis were seen in the ventricular pressure. Maximum mitral valve opening area was fixed at 2.0cm2 and mitral regurgitant area was 0.01 cm’. Aortic regurgitant area was also fixed at 0.01 cm’. Peripheral resistance and eripheral compliance were set at their ‘standard’ va Pues of 1.08 mmHg ml-’ s-r and 1.5 ml mmHg-’ respectively’. Both valves o ened to maximum orifice over a period of 50ms. & e time step used for each iteration, as in all the examples discussed below, was 5ms.
Figure 5 shows some representative examples of ventricular pressure plotted against stenotic valve orifice area in the range 0.5-2.0cm2. Results for heart rates of 40 beatsmin-‘, 80 beatsmin-’ and 120 beatsmin-’ are shown for stroke volumes of 40 and 100ml. Isolated aortic regurgitation This finding, too, is not uncommon with both native valves and bioprosthetic pericardial valves. The above set of heart-rate/stroke-volume simulations was repeated, only this time maximum aortic area was fixed at 2.0cm2 and the regurgitant area was varied over the range 0.01-o. 15 cm2. Figure 6 shows the effect of increasing regurgitant area on the three pressures: mean ventricular pressure over s stole, systolic arterial pressure and diastolic arteria r ressure. Figure 7 shows how the regurgitant fraction Ptotal regurgitant-volume/stroke-volume, expressed as a percentage) varies with regurgitant aortic area for
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Regurgitant area (cm2 ) arterial 6 Mean ventricular pressure over systole (-), systolic pressure (0 -Cl) and arterial diastolic pressure (A-A) plotted against aortic valve regurgitant area for a heart rate (HR) of 60 beats min-’ and stroke volume (SY) of 80 ml
Figure
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Figure 8 Factor by which the work performed by the heart to deliver a givenvolume to the peripheral circulation must be increased in the ), 40ml at 40beatsmiu’; presence of aortic regurgitation. ((A-A), 60ml at 80beatsmin’; (O-Cl), 80ml at 120beatsmiu
rate/stroke-volume calculated from:
combinations.
The vertical axis is
Excess work factor = (SV& x Pj&../(sVo
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where SV,, and Vpregare the required stroke volume pressure in the and mean systolic ventricular presence of regurgitation and SF’,, and VP, are the stroke volume and mean systolic ventricular pressure when there is no regurgitation.
Limiting of cardiac output in isolated aortic stenosis
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Figure 7 Regurgitant fraction plotted against aortic valve regurgitant ), 80 (A-A) and 120 beatsmin-’ area at heart rates (HR) of 40 ((O-Cl) for a stroke volume of 80 ml
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Cardiac adaptation regurgitation
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Most forms of valvular disease develop over a prolonged period and there is usually concomitant cardiac adaptation. In the simulation above, the stroke volume remained fixed and the effect of isolated regurgitation on ventricular and arterial ressures was investigated. In the real situation, the g ear-t would ada t, insofar as it was able, to deliver the net volume Kow re uired by metabolic demands. The model was adapte 8 to simulate this situation for different heart rates and net stroke delivered to the peripheral circulation (s stohc stroke-regurgitant volume). The effect of the CT ifferent levels of regurgitation is probably best demonstrated by deriving the factor by which the work done by the heart must be increased to deliver the same volume as would be delivered in the absence of regurgitation. The results are illustrated in Figure 8 for three different heart-
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Just as the heart adapts in the presence of aortic regurgitation, it adapts to aortic stenosis. It would be interesting to know the maximum cardiac out ut that can be attained in the presence of a given Peve1 of stenosis. An approximate answer to this uestion may be obtained by assuming an upper ‘fimit to the maximum ventricular pressure that can be sustained. Having set this pressure ceiling, the maximum stroke volume which can be delivered for a given stenotic
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Figure 9 Maximum attainable stroke volume (for mean ventricular pressure of 200mmHg) for different grades of stenotic aortic orifice. ), 120 beats min-‘; (A - A), 80 beatsmin-’ (-
Circulatory qtem
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DISCUSSION The major
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Figure 10 Plots illustrating the linear relationship between mean ventricular pressure (), arterial systolic pressure (A -A) and arterial diastolic pressure (O--O) and peripheral resistance. Results shown are for a stroke volume of 100 ml at 60 beats min-’ without any aortic regurgitation
area, for a given heart rate, can then be calculated. In the examples illustrated in Figure 9, the maximum sustainable mean ventricular pressure was set at 200mmHg. This is an arbitrary figure, but a reasonable one for the purpose of illustration. Maximum stroke volume is plotted against re gitant area for two different heart rates, 80 Yreatsmin-’ and 120 beats min-‘.
Variation of peripheralresistance Problems of the circulation are not always related solely to the heart and heart valves. There are often associated changes to the systemic arteries. S stemic variations may be modelled by varying ei x er the peripheral resistance R, or the compliance C, or both. Figure 70 shows the variation of mean systolic ventricular, systolic arterial, and diastolic arterial pressures with peripheral resistance for a stroke volume of 100 ml at 60 beats mint. Peripheral resistance ranges from 0.7 mmHgmll’ s-‘, which is ‘hypotensive’, to 1.5 mmHg ml-’ s-l, which is ‘hypertensive’. Figure 17 demonstrates the effect of peripheral resistance on the regurgitant volume in the presence of different levels of regurgitation. The stroke volume in these examples
40
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Figure 11 Regurgitant volume plotted against peripheral resistance for two values of regurgitant aortic area (& Stroke volume in all 80 (A - A) and 120 cases was 80m1, heart rates were 40 (O-Cl) beatsmin~’
effect of isolated aortic stenoses is to increase the ventricular pressure during systole, Figure 5. The increase in ventricular pressure becomes dramatic when the orifice area falls below 1.0 cm2. Even at a heart rate of only 40 beats min-‘, the mean ventricular pressure required to maintain a stroke volume of lOOm1 when the aortic orifice is 0.5 cm2 rises to over 440mmHg. Such a ressure is extremely unlikely to be achieved in vivo. ?1%us, stenosis imposes a limit on the volume that can be delivered by the heart with each stroke. This is illustrated by the results of Figure 9. A maximum sustainable mean systolic ventricular pressure of 200 mmHg has been assumed in generating the two curves, for heart rates of 80 and 120 beatsmin-‘. This figure illustrates clearly the limiting effects of severe aortic stenosis on the cardiac output. For example, at 80 beatsmin-’ the maximum stroke volume that can be delivered for a stenotic orifice area of 1 cm2 is 75 ml, which gives a maximum cardiac output of 6.0litremin-‘, enough to maintain an individual at rest but not allowing much scope for exercise. The effects of acute isolated aortic regurgitation, such as might occur with leaflet tear, are examined in Figures 6 and 7. Ventricular and aortic systolic and diastolic pressures fall off almost linearly with the re rigtant area. There are differences in detail with di $ferent heart rate and stroke volume combinations, but the overall effect is the same, an almost linear falloff in all three pressures. Figure 7 is interesting because it combines the effects of different heart rates and regurgitant areas on the regurgitant fraction (percentage of stroke volume which flows back through the regurgitant valve during diastole). At 40 and 80bpm, the regurgitant fraction is dominated by leakage during the diastolic interval, the former bein worse because of the prolonged diastolic interva 4i. At 120beatsmin’, the pressure in the arterial s stem rises more sharply than at the lower rates. TZ us, although the diastolic interval as a proportion of cycle length is shorter at 120 beatsmin-* than that at 80 beatsmin-*, the increase in pressure means that the regurgitant fraction is actually higher. The volume swept back during valve closure is also increased at higher pressures. This explains why the curves for 120 and 80 beats min-’ in Figure 7 do not project back to zero for zero regurgitant area. In most cases, the heart will adapt to the regurgitant situation to deliver the necessary volume flow. To do so, it must work harder. The factor b which the workload on the ventricle is increased ?Jy regurgitation is illustrated in Figure 8 for three different heartrate/stroke-volume combinations. Not unexpectedly, the situation is worst at the lowest heart rates. For example, at 40 beatsmin-’ with a regur ‘tant area of only 0.15 cm2 the ventricle must wor !Y 2.66 times harder per beat to deliver a stroke volume of 40ml. The excess work factor is improved by increasing heart rate because the diastolic interval available for regurgitation is shortened.
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However, expressing the results in this way belies the true situation, particularly as the stroke volume increases. As heart rate increases for a given stroke volume the pressure in the arterial system inevitably rises and the ventricle has to work harder to overcome this higher pressure. Thus, the baseline work, in the absence of regurgitation, will be higher and although the factor by which the work must be increased to cope with regurgitation is less, the ventricle may not be able to reach the workloads demanded. This is illustrated for the situation represented by the point in Figure 8 corresponding to a stroke volume of 80 ml at 120 beats min-’ with a regurgitant area of 0.15 cm’. The increased workload is only a factor of 1.39 higher et the mean ventricular ressure required is 236mm ?I g. Volume delivery is Pikel to be compromised in this situation. These resu rts assume a maximum aortic orifice of 2.0 cm’. The situation would be made worse by coexisting aortic stenosis. Figure 70 illustrates the almost linear relationship between the pressures within the system and the peripheral resistance. Under the severe hypotension represented by a peripheral resistance of the systolic pressure falls to 0.7mmHgmll’s-l, 97mmHg and the diastolic pressure drops to only 47 mmHg, pressures which would be barely adequate to support the circulation. In h ertension, produced to increasing periphera Yp resistance bY 1.5mmHgmlr s-r, systolic and diastolic pressures and the heart rise to 175 and 125 mmHg, respective1 must work harder pro ortionately to (Yeliver the flow required by the perip Kera1 tissues. In the presence of a regurgitant area, the effect of increasing peripheral resistance is to increase the regurgitant volume, Figure 17. There are two interesting points to note about this figure. As was observed when considering the acute effects of regurgitation in Figure 7, regurgitation has its worst effects when the heart rate is low and the diastolic interval, correspondingly, is long. At higher heart rates, the pressure in the arterial system rises. Thus, in certain circumstances, regurgitation is higher at a rate of 120 beats min-’ than at 80 beats min-’ though the diastolic interval, as a proportion of the cycle length, is less. However, this does not hold in all situations. For example, as the area available for regurgitation increases in tandem with an increase in peripheral
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resistance, the 120 beats min-’ and the 80 beatsmin-’ plots cross over and the regurgitant volume at 80 beatsmin-’ becomes greater than that at This may be explained by the 120 beats min-‘. sh er fall-off in pressure during diastole as the area avai “p able for regurgitation increases. There seems to be no similar work with which to compare the model discussed here. The simplifying assumptions outlined in the description of the model limit its applicability as a simulation of the situation in vim. Its greatest value is in helping to understand the complex interaction between heart valves and the rest of the circulation. Valve opening action is modelled simplistically and variation of systolic and diastolic intervals with heart rate is not taken from an ‘exact’ physiological relationship, though it is doubtful that such a thing exists. On the other hand, the approximations made are not so far removed from reality as to be unreasonable. In fact, in our heart valve test laboratory, features predicted by the model have been observed subsequently in tests on prosthetic valves in vitro. The major limitation of the model is that it ignores the pressure wave and wave reflections and is, therefore, unable to predict the exact shape of the ressure profiles. Conversely, the prediction of flow Kehaviour, including closing and leakage regurgitation through defective valves, is a reasonable approximation of that observed in our laboratory studies of prosthetic heart valves. Features such as variation of systemic compliance or resistance with distending pressure would be relatively easy to build into the model. Further realism could be added by measuring valve opening areas directly frame-by-frame from high speed tine or video images (in vitro situation only) and thereby developing a better model of valve opening action.
REFERENCES Frank 0. Schltzung des Schlagvolumens des menschlichen Herzen auf Grund der Wellen und Windkessels theorie. 2 Biol 1930; 90: 405-9. Randall JE. Microcomputers and physiological simulation, New York: Addison-Wesley, 1980; chapter 9: 116-28. Shortland AP, Cochrane T. Doppler spectral waveform generation in vitro: an aid to diagnosis of vascular disease. Ultrasound Med G? Biol 1989;
15: 737-48.
