Hemodynamics of the ventricular outflow tract

1. Biomrcknicr.

Vol. 3. pp. 13-129.

Pergamon Press. 1970.

Printedin

Great Britain

HEMODYNAMICS OF THE VENTRICULAR OUTFLOW TRACT”? JOSEPH P. ARCHIE. JR. Department of Surgery. University of Alabama Medical Center. Birmingham. Alabama35233. U.S.A. Abstract-The purpose of this report is to formulate from the equations of fluid mechanics. assuming inviscid flow of an incompressible fluid. an equation for the fluid dynamics of the ventricular outflow tract. The unsteady flow term of the equation implies the possibility of a negative pressure gradient from the left ventricle to the proximal aortic arch in the latter half of systole. A numerical computation based on experimental data of others verifies this observation in the left ventricle-proximal aortic arch of dogs. The unsteady flow term gives new incentive to the use of ventricular hydrostatic pressure in the definition of total vascular impedance and ventricular work and energy.

IT HAS been shown by Spencer and Greiss (1962) that the hydrostatic pressure gradient between the left ventricular chamber and proximal aortic arch may become negative in late systole. This means that the aortic pressure is greater than ventricular pressure during late outflow. Noble (lP68) has demonstrated conclusively in the dog using high fidelity micromanometers that this pressure gradient becomes negative after 30-50 per cent of the ejection period. He proposed a teleologically interesting explanation in terms of the blood momentum, but to date no fundamental explanation of this paradox has been given. This report utilizes the basic equations of fluid mechanics applied to the flow of the ventricular outflow tract to prove theoretically that this phenomenon is an expected occurrence. THEORY

In order to extract useful information from theory a multitude of simplifying assumptions must be made in any hemodynamic problem. The principal assumptions used here are the inviscid flow of an incompressible fluid. As blood is essentially incompressible over the range

of pressures physiologically applied, the latter assumption is well accepted. The former, that of inviscid flow, requires some comment. To treat the blood in the outflow tract of the left ventricle mathematically as a viscous fluid (which it is) brings on a great many complications not only in the inclusion of the viscous terms of the Navier-Stokes equations but the fact that turbulence, the development of boundary layers, and non-Newtonian effects are probably present to some degree. If the model is assumed to be inviscid unsteady flow. the viscous terms do not appear in the equations of fluid mechanics and the effects of the unsteady component of the flow can be examined after integration of the equations. The unsteady viscous form of the equations is not integrable in the context of our present knowledge of differential equations. It is the unsteady flow terms of the motional equations that yield the solution and hence explanation as to why the above defined pressure gradient becomes reversed in latter systole. The procedure employed is the integration of the inviscid unsteady Navier-Stokes equation (Euler equations) on a streamline as

*Received 5 Januay 1970. tGrand Award. SAMA-Mead Johnson Research Forum 1969. Read in part before the scientific sessions of the AMA Annual Convention. New York, 1969.

426

J. P. ARCHIE,

shown in Fig. 1. Under steady flow conditions and the above assumptions the result would be the well-known Bemouilli equation

where 1 and 2 are the arbitrarily selected initial and final points on a streamline, V is

v. t

JR.

velocity and pressure are written as functions of time in equation (2). A crude. but sometimes very practical. technique to include the effects of fluid viscosity in energy dissipation is effected by addition of an energy loss term to the left side of equation (2) of the form IV,_.,. Eshinazi (1962). If point 1 is in the left ventricle at a zero flow position V, = 0, the pressure gradient becomes 4p = p, -py = pVr2/7+pI; +pw,+

(aV/ar)ds (3)

where point 2 is in the proximal aortic arch. This equation states that the pressure gradient from ventricular chamber to aortic arch is equal to the kinetic energy in the arch at point 2. plus the contribution due to unsteady flow. plus the energy per unit mass dissipated because of viscosity between points 1 and 2. DISCUSSIOS I

I

I

I

I

I

I

2

Fig. 1. A streamline in the ventricular outflow tract where point I is in a zero flow region in the ventricle and point 2 in the proximal aortic arch.

the velocity vector. p is the hydrostatic pressure, and p is the fluid density. a constant for incompressible fluids. In the unsteady case a modified Bemouilli equation is obtained V,P(f)/2+pl(r)/p

Figure 2 is a typical flow curve for systolic ejection from the left ventricle. The spatial average velocity. V. would be similar as Q = A I/, where Q is flow and A is the crosssectional area. It is readily apparent from equation (3) that the only possible contribution to 4p could result in a negative value of 4p is the unsteady flow term J: (aV/‘lar)ds. Figure 2 clearly demonstrates that this term will reverse sign when t > c’ as the derivative

= V,‘(t)/2+pr(t)/p+ + J; (aI”(s.r)/at)

ds.

(2)

In the above equations the identical terms are the kinetic energy and pressure energy at points 1 and 2. Equation (2) contains an additional term which arises because of unsteady flow (i.e. the velocity is not a constant with respect to time). This term is an integral along the streamline of length s from points l-2. The

Fig. 2. A typical ejection flow curve where t,, is the onset time of ejection. t, the time of completion. and r”the time of maximum flow Q and velocity V.

THE VENTRICULAR

av/at becomes negative. The relative magnitude of the three terms on the right side of equation (3) will determine for each instant of time the numerical value of Ap( t). The energy loss term W,_, is small compared to Ap under physiological conditions and constitutes perhaps at most l-2 mm Hg or less pressure drop on the average. It would, however, be a major determinant of Ap in pathologic states characterized by high output or by obstruction to flow such as aortic stenosis, as implied by the application of equation (3) to the data of Spencer and Greiss (1962). The always positive term Vzz/2 would be maximum at t’ and decrease in magnitude for t, > t > f’. The derivative aV/dr. however, is zero at r’ and increases negatively in early t > t’ and remains negative for all t1 > t > t’. Therefore, the possibility of a negative Ap during some period t > t’ exists under physiologic conditions. Noble (1968) clearly points out that the momentum of decelerating fluid is responsible for the negative pressure gradient. The formulation presented in this report is not in terms of momentum as such but arises in a natural way from integration of the equations of conservation of linear momentum. in classical mechanics called Newton’s second law of motion. It should be made clear that the presence of a hydrostatic pressure in the proximal aortic arch numerically greater than that in the left ventricle during systole is not a reason to alter our thinking about left ventricular function. The pressure gradient is merely a state of dynamic equilibrium. This point can be illustrated by steady flow of an incompresible fluid in tubes. If the fluid flows from a section of a given cross-sectional area to one of larger area the hydrostatic (lateral) pressure increases provided there is in reality minimal energy dissipation. The downstream hydrostatic pressure is greater than that in the smaller cross-sectional area section because the kinetic energy per unit mass contained in the fluid is transported to pressure energy as it

427

OUTFLOW TRACT

slows. This is predicted by the Bemouilli principle, can be verified experimentally. and constitutes a state of dynamic equilibrium. The interpretation of Noble (1968) that in late systole ejection results from momentum of blood with minimal myocardial contraction is in part true. The momentum per unit mass given during the rapid ejection phase when aV/at is positive is transported back to the fluid during the negative aV/dt phase and appears as an energy increase in p.,(t) and more distally perhaps in part as elastic energy in the vessel wall. The ventricle performed this increment of work in early systole. It is not returned to the ventricle but rather stored in a different form by momentum transfer. The work done over any time period is given by W(t) = /p, (1) Q(r) dr

(4)

where Q is the flow and the limits of integration are set by the time interval desired. The energy expended at any time is E = dWldr = p1 (I) Q(r).

(5)

As Q is positive throughout systole. energy is expended throughout and work is done by the ventricle. It may appear from examination of pressure alone that there is some question about the contribution of the ventricle to late ejection. The definition of work and energy given above is consistent with active contraction to some degree in late systole as it continues to do work at high pI. It is true that more work is done during early systole than late systole at a given flow because of the rate of change of momentum. This greater energy expenditure in early rather than in later systole at the same flow is reflected through the larger value of p1 early. NUMERICAL

EXAMPLE

For purposes of computation the velocity vector P7t.s) may be separated into spatial (along streamline) and temporal components,

128

J. P. ARCHIE. JR.

V= V(s,f) =A(s)B(t). where A(s) is anondimensional function of maximum amplitude of unity. The term A(s) could be represented by an infinite series of polynomials in s. B(t) is the temporal component of a pulsatile nature easily represented by a Fourier series. Figure 1 is a schematic of the ventricular outflow tract. The boundary condition is V(o, t) = 0. that is, when s=O (at point 1) there is no flow. The initial and final conditions are Vs, to) = 0 and V/(s, tl) = 0 where r0 is the time of ejection onset and t, is the time at completion. If, for simplicity. A(s) = a + bs where a and b are constants, then a = 0 and b = l/L where L = J,? ds. the length of the streamline. Then Y(s, t) = (s/L)B(t) and V(L, t) = V2(2)= B(t). Equation (3) becomes Ap=p1--pz=

loo

.

x

9

E

0’

;

1

PWI-_?.+(pL/2)~~+(p/2)V32 (6)

for t1 zz t b r,,, where i/ = dMdt. To effect a computation demonstrating the possibility of a negative Ap the following values are introduced from Spencer and Greiss (1962) and O’Rourke (1968); V,, = 130 cmlsec, L=lOcm, t=t,-r,=O.l5sec, and a first term approximation of V(L, t) of the form (Fig. 3(a)) Vz(t) = paX sin &/(t, -to). The results of the computations are displayed in Fig. 3(b) where the value of W,+ has been neglected. The maximum and minimum value of Ap is plus and minus 10.8 mm Hg respectively using proper conversion units. The numerical results of equation (6) depend on the value of L which was not measured but herein estimated. For various L values a family of solutions is obtained as presented in Fig. 3(c). Noble (1968) obtained negative pressure gradients of 5-20 mm Hg. If a true flow or velocity profile were used with peak spatial average velocity V in early systole as in Fig. 2. the shift to a negative pressure gradient would come earlier, as was obtained experimentally by Spencer and Greiss (1962) and Noble (1968). The value of W,_? acts as a bias on the

I

0

I

0.5

I

t/~t,-to)

Fig. 3. Part (a) is the assumed velocity function cos rrf/(f,-I,), (b) is the solution to equation (6). and (c) demonstrates the numerical solution’s dependence of L. value of Ap, adding a constant positive value to it. If the full Navier-Stokes equations were solvable the contribution of viscosity would enter as a time dependent parameter over systole with an average value of W,_,. The effects of elastic wall energy storage have been neglected herein because point 2 is considered just distal to the aortic valve. CONCLUSION This report is purely theoretical and apart from offering a plausible explanation for the observed events~~ of others.’ it serves as a _ __ .~

THE \‘ESTRICULAR

rational way to look at flow in the ventricular outflow tract. The addition of a new term to the energy balance across this region brings up interesting questions in terms of the definition of total vascular impedance and ventricular energy. O’Rourke (1968) has pointed out that the total or impact pressure (lateral plus kinetic pressure) in the proximal aortic arch should be used rather than the lateral pressure. It is suggested here in that the systolic ventricular pressure be used for total vascular impedance as it represents the total energy lost in the vasculature, it either being measured or computed by equation (3). Furthermore, the ventricular pressure. p,. should be used to compute ventricular work and energy. REFERENCES Eskinazi, S. (1962) Principles ofFhid 5. Allyn and Bacon, Boston.

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Noble. M. L. M. (1968) The contribution of blood momentum to left ventricular ejection in the dog. Circdnrion Res. 23,663-670. O’Rourke. M. F. (1968) impact pressure. lateral pressure. and impedance in the proximal aorta and pulmonary artery. J. nppl. Physiol. X,533-53?. Spencer. M. P. and Greiss. F. C. (1961) Dynamics of ventricular ejection. Circulation Res. 10,274-278.

NO>lENCLATURE velocity. cmlsec p pressure. mm Hg p density t time. set a partial derivative .f integral W energy loss Q flow rate A area E energy V

9 =

Mechnnics,

TRACT

Chap. L

c!

dr outflow length. cm