Fluid Dynamics of Prosthetic Valves

Acknowledgments

The work on prosthetic heart valves was performed in our Cardiovascular Fluid Mechanics Laboratory at Georgia Tech University and was supported by grants from the US Food and Drug Administration, the American Heart Association, the heart valve industry, and generous gifts from Tom and Shirley Gurley and the Lilla Boz Foundation.

Doppler echocardiography has made noninvasive examination of prosthetic heart valve function a clinical reality. Unfortunately, there are still many imperfections in echocardiographic examinations due to acoustic shadowing, reflections caused by implanted valves, and limitations in ultrasound technology. Nevertheless, a number of important parameters can be calculated or estimated to aid in the assessment of prosthetic heart valve function and performance.

After a valve has been implanted, its function is governed primarily by its hemodynamic characteristics. To understand the hemodynamic performance of prosthetic heart valves, it is necessary to have a solid background in the physical laws that govern their function. This chapter introduces cardiologists to the governing principles of fluid dynamics, relevant formulations used in prosthetic valve assessment, and fluid mechanical characteristics of specific prosthetic heart valves.

Principles of Fluid Dynamics

Conservation of Mass

The principle of conservation of mass is important in evaluating valve effective orifice area (EOA). Conservation of mass, or the continuity principle, is a mass balance over the boundaries of a volume of fluid. For example, a volume can be made to coincide with a section of an artery, with the left ventricle (LV), or with the surface of a blood cell, and it also can be made to move over time. This is the control volume , and its boundaries are usually chosen to give information regarding an unknown flow rate, average velocity, or surface area flow into or out of the volume based on more easily measurable quantities. Examples of control volumes useful for determining aortic and mitral valve areas with conservation of mass are shown in Fig. 30.1 .

Fig. 30.1, Control volume analysis to determine valve area.

Conservation of mass within the control volume can be expressed as follows:

\begin{matrix} Rate of mass accumulation = \\ Rate of mass input - Rate of mass output \end{matrix}

$\begin{matrix} Rate of mass accumulation = \\ Rate of mass input - Rate of mass output \end{matrix}$

In cardiovascular applications, we can assume that the density of blood is constant because blood is composed mostly of water, which is almost incompressible, and thermal expansion or contraction is insignificant due to the narrow range of body temperature. Mass is the product of density and volume, and mass conservation can be expressed as volume conservation:

\begin{matrix} Rate of volume accumulation = \\ Rate of volume input - Rate of volume output \end{matrix}

$\begin{matrix} Rate of volume accumulation = \\ Rate of volume input - Rate of volume output \end{matrix}$

In mathematical form, this is

\frac{(V_{f} - V_{i})}{t} = Q_{i n} - Q_{o u t}

$\frac{(V_{f} - V_{i})}{t} = Q_{i n} - Q_{o u t}$

where Q _in is the total flow rate into the control volume, Q _out is the total flow rate out of the control volume, V _i is the size of the control volume at the beginning of the observation, and V _f is the size of the control volume after an observation at time t . There can be multiple sources of flow inlets and outlets:

\frac{1}{t} (V_{f} - V_{i}) = (Q_{i n (1)} + Q_{i n (2)} + Q_{i n (3)}) - (Q_{o u t (1)} + Q_{o u t (2)} + Q_{o u t (3)})

$\frac{1}{t} (V_{f} - V_{i}) = (Q_{i n (1)} + Q_{i n (2)} + Q_{i n (3)}) - (Q_{o u t (1)} + Q_{o u t (2)} + Q_{o u t (3)})$

If we further assume that the volume of the control volume does not change with time, the left-hand side of Eq. 30.3 is zero, and conservation of mass can be expressed as follows:

Q_{i n} = Q_{o u t}

$Q_{i n} = Q_{o u t}$

This is approximately true in most blood vessels if we neglect expansion vessels under elevated pressure. Often, flow rate in circular, nonbranching vessels can be decomposed into an average axial velocity ({vbar}) and a cross-sectional area (A) . Eq. 30.5 becomes

{\bar{v}}_{i n} A_{i n} = {\bar{v}}_{o u t} A_{o u t}

${\bar{v}}_{i n} A_{i n} = {\bar{v}}_{o u t} A_{o u t}$

Flow through a prosthetic aortic valve is an example of this situation. The rate of fluid mass moving toward the valve from upstream has to equal the rate of fluid mass moving through the vena contracta or further downstream of the aorta. This principle is the basis of the equations for calculating EOA using Doppler echocardiography.

Mechanical Energy

Mechanical energy can be described as the ability to accelerate a mass of material over a certain distance. Energy per unit volume has the same dimensions as pressure. It is perhaps best to report energy in cardiovascular applications on a per-unit-volume basis because most members of the medical field are accustomed to working with units of pressure. However, it is important to consider pressure as only one of several forms of mechanical energy.

Pressure (p) can be shown to be a form of mechanical energy per unit volume by recognizing that it represents a force (F _p ) per unit area (A) . If a force is applied to move a mass over a distance ( d ), it performs work, expending pressure energy (E _p ) . The area that the force acts on multiplied by the distance over which it moves represents a volume (V) :

\frac{E_{p}}{V} = \frac{F_{p} d}{A d} = \frac{F_{p}}{A} = p

$\frac{E_{p}}{V} = \frac{F_{p} d}{A d} = \frac{F_{p}}{A} = p$

Pressure is not the only form of mechanical energy in the circulation. Acceleration due to gravity (g) creates another form of mechanical energy per unit volume. Gravity can accelerate a mass (m) with a force (F _g ) . If this force moves the mass over a vertical distance (h) , it too performs work and expends gravitational energy (E _g ) . The energy per unit volume is obtained by substituting density ( ρ ) for mass per unit volume.

\frac{E_{g}}{V} = \frac{F_{g} h}{V} = \frac{m g h}{V} = ρ g h

$\frac{E_{g}}{V} = \frac{F_{g} h}{V} = \frac{m g h}{V} = ρ g h$

Mechanical energy per unit volume exists in the circulation in the form of kinetic energy, or energy of movement. If a mass (m) is moving at a velocity (v) , it contains kinetic energy, equivalent to half the product of the mass and the square of the velocity:

\frac{E_{k}}{V} = \frac{\frac{1}{2} m v^{2}}{V} = \frac{1}{2} ρ v^{2}

$\frac{E_{k}}{V} = \frac{\frac{1}{2} m v^{2}}{V} = \frac{1}{2} ρ v^{2}$

The energy per unit volume is obtained by substituting density for mass per unit volume. There are three forms of mechanical energy in the circulation.

The primary source of mechanical energy in the circulation is the work performed by the LV. Contraction of the ventricle creates an increase in pressure inside the ventricle. As Eq. 30.7 implies, subsequent movement of a volume of fluid by this pressure results in the generation of mechanical energy. The energy generated by the ventricle during one cardiac cycle is illustrated in Fig. 30.2 . This diagram represents the pressure and volume within the ventricle during the course of one cardiac cycle. The curve between points 1 and 2 represents the period of diastolic filling, when the mitral valve is open and the ventricle fills. The mitral valve shuts at point 2, and the period of isovolumic contraction begins. Between points 2 and 3, the volume of blood in the ventricle remains constant, but the pressure rises considerably. The aortic valve opens at point 3. This starts the period of systolic ejection, during which blood is moved by the ventricle into the aorta. At point 4, the aortic valve closes and the period of isovolumic relaxation begins. During this period, the ventricular volume remains constant, but the pressure falls, returning the ventricle to its state at point 1.

The energy generated by the ventricle during one cardiac cycle is equivalent to the integral of the pressure-volume diagram (the shaded area in Fig. 30.2 ). The energy per unit volume generated by the ventricle is equivalent to the integral of the pressure with respect to volume divided by the stroke volume. This is roughly equivalent to the average increase in LV pressure from diastole to systole. The ventricle creates energy in the form of pressure, but this energy is converted to gravitational and kinetic energy elsewhere in the circulation.

Bernoulli’s Equation

Pressure, gravitational, and kinetic energies in the circulation can be freely converted from one to another without energy loss. Bernoulli’s equation for steady flow illustrates this. The Bernoulli equation relates the relative amounts of pressure, gravitational energy, and kinetic energy per unit volume between two spatial locations along the path of a flow (locations 1 and 2, where location 2 is downstream of location 1), assuming that no energy is lost:

\frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} + p_{1} = \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2} + p_{2}

$\frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} + p_{1} = \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2} + p_{2}$

The Bernoulli equation for steady flow states that the total mechanical energy per unit volume at locations 1 and 2 are the same but can exist in different forms. It shows that a decrease in pressure from location 1 to location 2 may be balanced by an increase in fluid velocity or height without loss of energy. A pressure drop is not mechanical energy loss if it is accompanied by an increase in gravitational or kinetic energy; these energies can be converted back to pressure energy later.

Additional variables may be added to the equation to account for the effects of unsteady flow and mechanical energy loss:

\frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} + p_{1} = \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2} + p_{2} + \int_{1}^{2} ρ \frac{\partial v}{\partial t} \partial s + Φ

$\frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} + p_{1} = \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2} + p_{2} + \int_{1}^{2} ρ \frac{\partial v}{\partial t} \partial s + Φ$

where s represents the distance of the path between locations 1 and 2, and Φ represents loss of mechanical energy per unit volume. The added terms represent the contribution of acceleration of the fluid to flow energy and the conversion of mechanical energy to heat, respectively, between locations 1 and 2. Rearranging Eq. 30.11 to express energy loss in terms of the other parameters yields the following:

Φ = (p_{1} - p_{2}) + \frac{1}{2} ρ (v_{1}^{2} - v_{2}^{2}) + ρ g (h_{1} - h_{2}) - \int_{1}^{2} ρ \frac{\partial v}{\partial t} \partial s

$Φ = (p_{1} - p_{2}) + \frac{1}{2} ρ (v_{1}^{2} - v_{2}^{2}) + ρ g (h_{1} - h_{2}) - \int_{1}^{2} ρ \frac{\partial v}{\partial t} \partial s$

Control Volume Energy Analysis

A more detailed analysis of the energy of fluid flow through a valve can be conducted using the control volume approach. Bernoulli’s equation describes energy conversion and loss along a streamline (line of fluid flow), and it is a useful, simplified approach to analyze fluid energy in valves, whereas the control volume approach provides three-dimensional (3D) analysis of fluid energy at any time in the cardiac cycle. If the control volume is the blood volume around the valve, as shown in Fig. 30.3 , manipulation of the Navier-Stokes governing equations of fluid flow gives the following equation :

\int S_{i j} T_{i j} d V = \int - P (u_{i} \cdot {\overset{⌢}{n}}_{i}) d A - \int ρ u_{i} g_{i} d V - \frac{d}{d t} \int \frac{1}{2} ρ u_{i}^{2} d V

$\int S_{i j} T_{i j} d V = \int - P (u_{i} \cdot {\overset{⌢}{n}}_{i}) d A - \int ρ u_{i} g_{i} d V - \frac{d}{d t} \int \frac{1}{2} ρ u_{i}^{2} d V$

Fig. 30.3, Control volume analysis for the aortic valve.

where S is the shear stresses, T is the shear deformation, P is pressure, u is velocity, A is the surface of the control volume, n is the normal direction at the surface of the control volume, g is the gravitational force, and V is the control volume. The subscripts i and j refer to any of the three Cartesian coordinate axes, in accordance to Einstein’s notation. Eq. 30.13 can be described in words as follows:

\begin{matrix} Energy dissipation = Energy supplied by pressure gradient - \\ Energy converted to gravitational potential energy - \\ Energy converted into kinetic energy \end{matrix}

$\begin{matrix} Energy dissipation = Energy supplied by pressure gradient - \\ Energy converted to gravitational potential energy - \\ Energy converted into kinetic energy \end{matrix}$

The pressure gradient across the valve, which is supplied by the heart muscles, is partially converted into kinetic energy (used to accelerate fluid) and partially used to counter the effects of gravity. The remaining energy is lost as heat and sound.

Mechanisms of Mechanical Energy Loss

Mechanical energy can be converted to heat and sound through friction between blood volumes moving at different velocities and between blood and the vessel walls. Heat and sound generated through friction is not readily converted back to mechanical energy, and this energy is said to be lost to the cause of blood circulation. The frictional losses take one of two forms: viscous losses and turbulent losses. Additional energy can be effectively lost in the circulation due to valvular leakage. These forms of energy loss are described in the following sections.

Viscous Losses

As a result of frictional forces, fluid immediately adjacent to a solid boundary moves with the same velocity as the boundary. In the case of a stationary vessel, this means that fluid immediately adjacent to the vessel wall does not move, no matter how fast the surrounding flow is moving. As the distance increases from the solid boundary, the velocity may increase. This leads to differences in fluid velocity with respect to radial distance within the vessel ( Fig. 30.4 ).

Fluid viscosity, the tendency of the fluid components or molecules to stick to each other, creates friction between fluid components that are close to one another if they move at different velocities. This is the mechanism of viscous energy loss. Viscous losses typically increase with flow rate, and they decrease dramatically with increasing vessel radius. This is why vasodilators are effective in relieving LV workload.

Turbulent Losses

Turbulent losses usually do not occur in the circulation but can be greater in magnitude than viscous losses when they do. Turbulence is characterized by chaotic spatial and temporal fluctuations in the direction and magnitude of fluid velocity. It is a result of the inertia of the flow being too great for frictional forces to stabilize fluid movement or to dampen unsteady secondary motions. Chaotic spatial and temporal velocity fluctuations result in excessive mixing of fluid; individual tiny pockets of fluid experience very high shear stresses with other pockets of fluid everywhere in the flow. Consequently, there are high overall frictional energy losses, which is the essence of turbulent losses.

In a straight vessel, fluid inertia disturbances may be initiated by small irregularities, or roughness, in the surface of the vessel. In laminar flow, these small inertia disturbances are quickly dampened by viscous forces, and unsteady secondary flow and fluid mixing are minimal. In turbulent flow, the inertia changes become amplified because viscous forces are not strong enough to dampen them. The analogy is being tripped while walking versus running. It is harder to maintain balance when tripped during running than during walking because the inertia of the body’s motion is difficult to control.

The tendency toward turbulence in a fluid flow is determined by the ratio of inertial forces to frictional forces in the flow. The inertial force of a moving flow is the force required to bring the flow to rest; the frictional force of such a flow is created by viscous shear stress acting on solid surfaces. In vessel flow, this ratio is approximated by the Reynolds number:

N_{Re} = \frac{D \bar{v} ρ}{μ}

$N_{Re} = \frac{D \bar{v} ρ}{μ}$

where D is the inner diameter of the vessel, v– is the average velocity of the flow, ρ is the fluid density, and μ is the fluid viscosity.

A low Reynolds number results in laminar flow, whereas a high Reynolds number results in turbulent flow. The point of transition between the two states, the critical Reynolds number, varies for different flow systems. For flow through a straight pipe, the critical Reynolds number is approximately 2000, and flow becomes fully turbulent at a Reynolds number of approximately 6000.

Flow through small, circular orifices creates a phenomenon known as a free jet . For steady free jet flows, the critical Reynolds number is approximately 1000, with fully turbulent flow occurring at Reynolds numbers greater than 3000. ^, This transition point is lowered by sharp corners and bends in the flow or rough solid surfaces. For turbulence to occur, sufficient time must be allowed for the unsteadiness to amplify and grow. In pulsatile flows, if the frequency of pulses is too high, there may be insufficient time for turbulence to fully develop, making it harder to achieve turbulence. The critical Reynolds number increases.

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