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Applied BioFluid Mechanics - Lee Waite and Jerry Fine

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172 Chapter Six

twice the area of the mitral orifice, allowing for a large area of coaptation permitting effective valve sealing.

The mitral valve is shown in a medial view in Fig. 6.3. The valve has been dissected through the anterolateral commissure and laid open. In this view it can be seen that the anterior and posterior leaflets are not separate entities, but rather one continuous piece of tissue.

The chordae tendineae for both leaflets attach to the papillary muscle. Chordae tendineae consist of an inner core of collagen surrounded by loosely meshed elastin and collagen fibers with an outer layer of endothelial cells. There is an average of about 12 primary chordae tendineae attached to each papillary muscle. The anterolateral and the posteromedial papillary muscle attach to the ventricular wall and tether the mitral valve into place. The chordae tendineae prevent the mitral valve from inverting through the annular ring and into the atrium. That is, in normal physiologic situations they prevent mitral valve prolapse. Improper tethering results in prolapse and mitral regurgitation.

Stresses in mitral valve leaflets of up to 0.22 MPa have been estimated when the left ventricular pressure reaches up to 150 mmHg during ventricular systole. Mitral velocity flow curves show a peak in the curve during early filling. This is known as an E wave. Peak velocities are typically 50 to 80 cm/s at the mitral annulus. A second velocity peak occurs during atrial systole and is known as an A wave.

In healthy individuals, velocities in the A waves are lower than the velocities during the E wave. Diastolic filling of the ventricle shows a peak in flow during diastolic filling followed by a second peak during atrial systole.

6.4 Pressure Gradients across a Stenotic Heart Valve

To assess the seriousness of a heart valve stenosis, it would be helpful to know the area of the stenosed valve when open, or how much of the valve opening is being blocked by the stenosis. Gorlin and Gorlin (1951) developed an equation to empirically predict valve area based on the pressure drop across the valve and the flow rate through the valve. The published equations that were useful for the prediction of area in mitral and aortic valves are still used in cardiology today.

Harris and Robiolio (1999) write, “Although the ‘gold standard’ for assessment of the severity of mitral stenosis has long been the Gorlin formula derived from data obtained from cardiac catheterization, advances in echocardiography have largely supplanted this procedure. Nonetheless, cardiac catheterization remains a mainstay in the management of mitral stenosis.”

Mechanics of Heart Valves


6.4.1The Gorlin equation

We will begin by writing the Gorlin equation in terms of variables that are consistent with the other fluid mechanics notations in this textbook. The volume of flow passing through the aortic valve per beat Q is the cardiac output CO divided by the heart rate HR.

Flow rate per beat 5






The average flow rate through the aortic valve is the same amount divided by the ejection time per beat, TE.



Q 5 flow rate per beat 5 sHRdsTEd

Next, remember that the mean velocity of the flow V (average flow velocity across the flow cross section, through the valve) multiplied by the cross-sectional area AV is equal to the flow rate.



The Bernoulli equation is an equation based on the principle of conservation of energy, in the absence of viscous losses. In fact, there will be significant viscous losses through the heart valve, but the Gorlin equation is an empirical equation intended to account for those losses with an empirically derived valve coefficient. The Bernoulli equation is as follows:

P1 1



5 P2 1











If point 1 referred to in the equation is a reservoir with a very small velocity compared to the jet through the valve, the equation could be rewritten:

V2 5


2 rP



or, in terms of the volume flow rate:




Q 5 AV r


The above equation relates flow rate, pressure drop, and valve area in the ideal case, but in the case of a real valve, two nonideal characteristics have

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to be considered. Real blood flow is viscous flow and therefore Eq. (6.5) uses

a valve constant, CV, to represent the actual velocity through the valve.












V2 5 CV







In the second nonideal condition, a jet of blood flowing through the valve will contract to a smaller area than the total cross-sectional valve area. The area of contraction A depends on the valve design and can be designated here using the valve coefficient CC.



Combining Eqs. (6.3), (6.7), and (6.8), the Gorlin equation becomes:















The coefficients are measured for various orifices and it should be carefully noted that both the valve coefficient and the contraction coefficient are empirically derived coefficients which may not be constant under varying pressure and flow conditions as well as varying Reynolds numbers.

In fact, this equation or a modified form of the equation is often used clinically to estimate the effective area of a stenotic heart valve. To further simply its use clinically, the equation is often written in the following form:

AV 5








where AV is the area of the stenotic valve, CO is the cardiac output in cc/min, TE is the ejection time or the time per beat during which blood flows through the valve. HR is the heart rate in beats per minute, P is the mean pressure gradient over the ejection period in mmHg, and K is the Gorlin constant.

The Gorlin constant is proportional to CC, CV, and 1/ 2r and contains all the conversion factors necessary to account for a mathematically consistent set of units. The flow rate is equal to:

Q 5 CO




If the cardiac output is given in cubic centimeters per minute, the ejection period in seconds per beat, and the heart rate in beats per minute, then

Mechanics of Heart Valves


Eq. (6.11) yields a flow rate in cubic centimeters per second. It is important once again to note that the Gorlin equation is a useful concept, but that the Gorlin “constant” is not a constant but depends on the flow rate.

Therefore, the Gorlin value of the Gorlin constant K is sometimes given as 44.3 in aortic valves and 37.7 in mitral valves. The units associated with the Gorlin constant are as follows:



6.4.2Example problem—Gorlin equation

A patient has a cardiac output of 5 L/min. The systolic ejection period is 358 ms with a heart rate of 70 beats per minute. The mean aortic gradient as measured by echocardiography is 81 mmHg. Find the aortic valve area as estimated by the Gorlin equation. What is the average flow rate across the aortic valve during ejection?



AV 5 TE HR s44.3d2 P

where AV aortic valve area, cm2 CO cardiac output, mL/min TE time of ejection per beat, s HR heart rate, beats/min

mean pressure gradient left ventricular pressure aortic pressure, mmHg


AV 5 s0.358ds70ds44.3d 281 5 0.50 cm2

The average flow rate across the valve is 5000 mL/25 s 200 mL/s.

6.4.3Energy loss across a stenotic valve

Energy loss across a stenotic valve is influenced by more than the effective valve area predicted by the Gorlin equation. Garcia et al. (2000) suggest an approach that uses an energy method to more accurately estimate the severity of an aortic stenosis. The authors suggest an index that they refer to as the energy loss index. Although it is a measure of energy loss, the energy loss index yields a value with pressure units.

The Gorlin equation uses transvalvular pressure gradients to predict effective orifice areas. These parameters are commonly used

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clinically for the assessment of aortic stenoses. Some studies have emphasized the importance of taking into account the pressure recovery phenomenon that occurs downstream from the valve. In other words, the kinetic energy that is stored in the fluid before, and as, it “jets” through the vena contracta is converted to pressure as the fluid is decelerated.

The measurement of a pressure gradient using Doppler echocardiography is an indirect measurement based on the velocity gradient. Those indirect pressure gradients can differ significantly from the actual pressure gradients measured via a catheter. This phenomenon is often referred to as “pressure recovery” and can be significant in patients with moderate to severe aortic stenoses and with small aortic crosssectional areas.

From the Bernoulli equation, a term the authors describe as energy loss EL can be written by:

EL 5 PVC 2 PA 1 1 rVVC2

2 1 rVA2





where PVC pressure at the vena contracta PA pressure in the aorta

VVC velocity of blood in the vena contracta VA velocity in the aorta

Considering linear momentum along the flow direction as shown in Fig. 6.7, the momentum equation can be written as:

PVC AA 2 PA AA 5 2rV VC2 AVC 1 rVA2 AA


Combining Eq. (6.12), the Bernoulli equation between the vena contracta and the aorta, and Eq. (6.13), linear momentum between the vena






Figure 6.7 Schematic representation of flow through an aortic valve where V is the left ventricle, VC represents the vena contracta, and A represents the aorta. (Modified from Garcia et al., 2000.)

Mechanics of Heart Valves


contracta and the aorta, to eliminate pressures in the equation yields:







EL 5 2 arVVC2


b 1 a


VA2b 1 a








The continuity equation yields:






and alternatively VA2 5














Because measuring the area at the vena contracta is problematic, and because it will be convenient to use VV AV multiplied together as cardiac output, the energy loss equation can be rewritten as:









EL 5 2 aVVC 2 VV

AA b


Now the energy loss can be calculated in terms of variables that are readily measurable through transthoracic echocardiography, as shown in Eq. (6.17).





EL 5

2 aVVC 2

AA b


where r density

VVC blood velocity at the vena contracta CO cardiac output

AA cross-sectional area of the aorta

EL energy loss expressed in units of pressure

The unitless energy loss coefficient associated with this energy loss is defined by the term:

Energy loss coefficient 5



sAA 2 EOAd



An energy loss index that is less that 0.5 cm2/m2 should probably be considered a critical value. Avalue less than 0.5 indicates a serious stenosis.

Recall that the effective orifice area EOA is the area at the vena contracta which is derived from the continuity equation, and is shown in Eq. (6.19).





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6.4.4Example problem—energy loss method

The same patient from the previous example problem using the Gorlin equation has a cardiac output of 5 L/min. Aortic area 4.9 cm2. Density 1060 kg/m3. Velocity at the vena contracta 1.66 m/s. The mean aortic gradient as measured by echocardiography is 81 mmHg.

Find the “energy loss” in mmHg and the energy loss coefficient.




CO 2




kg N s2




5000/60 cms












EL 5


aVVC 2



b 51060







AA cm2








kg m




100 cm










5 EL 5 13 mmHg
















































































From Eq. (6.19) the effective orifice area is
















5000/60 cms


5 0.50 cm










1.66/100 cm/s






















Finally, from Eq. (6.18) the energy loss coefficient is:

Energy loss coefficient 5




5 0.46

sAA 2 EOAd

s4.9 2 0.5d




6.4.5Clinical features

(Lingappa and Farey, 2000)

Chordae tendineae rupture and papillary muscle paralysis can be consequences of a heart attack. This can lead to bulging of the valve, excessive backward leakage into the atria (regurgitation), and even valve prolapse. Valve prolapse is the condition under which the valve inverts backward into the atrium. Because of these valve problems, the ventricle does not fill efficiently. Significant further damage, and even death, can occur within the first 24 hours after a heart attack because of this problem.

6.5Prosthetic Mechanical Valves

Early mechanical prosthetic heart valves were ball valves. In 1952, Dr. Charles Hufnagel implanted the first ball and cage valve into a patient with aortic valve insufficiency. Although the design of

Mechanics of Heart Valves


mechanical prosthetic valves has progressed significantly over the past 50 years, ball and cage valves are still the valve of choice by some surgeons.

Björk-Shiley designed the first tilting disc, or single leaflet, valve around 1969. Those valves used a tilting disc occluder that was held in place by a retaining strut. However, the development of the bileaflet valve by St. Jude Medical in 1978 has led to the predominance of this type of bileaflet valve in the mechanical valve market.

One example of a recently approved mechanical bileaflet aortic valve is the On-X valve (Fig. 6.8) manufactured by Medical Carbon Research Institute inAustin, Texas. The On-X was approved by the Food and Drug Administration (FDA) for implant in humans in May 2001. The mitral valve version of the On-X was approved for use by the FDAin March 2002.

The On-X consists of a pyrolytic carbon cage, a Dacron sewing ring fixed by titanium rings, and a bileaflet pyrolytic carbon disc.

Pyrolytic carbon is a material used in artificial heart valves because of its mechanical durability and its characteristic ability not to generate blood clots. The material can be said to have low thrombogenicity. Thrombogenicity is the tendency of a material in contact with the blood to produce a blood clot. Dr. Jack Bokros and Dr. Vincent Gott discovered pyrolytic carbon in 1966. At that time, pyrolytic carbon was used to coat nuclear fuel particles for gas-cooled nuclear reactors. General Atomics initiated a development project headed by Dr. Bokros to develop a biomedical

Figure 6.8 On-X prosthetic made by MCRI, Austin, Texas (a wholly owned subsidiary of Medical Carbon Research Institute Deutschland GmBH, Hanover, Germany; used with permission).

180 Chapter Six

grade of pyrolytic carbon. The result was Pyrolite, which was incorporated into heart valve designs and remains the most widely used material for heart valves today.

6.5.1 Clinical feature—performance of the On-X valve

Between December 2000 and January 2003, Özyurda et al. conducted a study in Turkey to assess the performance of the On-X valve (Medical Carbon Research Institute, Austin, TX). The authors wrote, “Few data are available on the clinical outcome of patients undergoing valve replacement with the On-X valve. Therefore, the aim of this study was to evaluate the early and midterm performance of this new generation bileaflet prosthesis.”

On-X valves were implanted into 400 patients. Of these 400, 120 had an aortic valve replacement (AVR), 258 had a mitral valve replacement (MVR), and 22 had a combined aortic and mitral valve replacement (DVR).

Patients were followed up prospectively at 3- to 6-month intervals. The overall hospital mortality in the study was 3.5 percent. When examining the patients 4 years after the initial surgery, the overall survival rate was 92 1 percent. Freedom from thromboembolism and thrombosis events at 4 years in the study were as follows:


99.1 percent for aortic valve replacement

98.3 percent for mitral valve replacement

94.7 percent for dual valve replacement patients


100 percent for aortic valve replacements

99.2 percent for mitral valve replacements

94.7 percent for dual valve replacements

The authors also wanted to establish data concerning pressure drops across the On-X. They performed an echocardiograph within 1 year of the aortic valve replacement and found that the mean aortic valve gradient was 12.86, 10.3 3, 9.0 4, 8.3 3, and 6.2 3 mmHg for 19, 21, 23, 25, 27/29 mm valve sizes, respectively. The mitral valve replacement mean gradient was 4.9 2, 4.5 1.2, and 4.0 0.8 mmHg for 25, 27/29, 31/33 mm valve sizes, respectively.

The conclusion of the authors after this study was that the “On-X valve is a highly effective mechanical valve substitute with low morbidity and mortality and good functional results.”

6.5.2 Case study—the Björk-Shiley convexo-concave heart valve

The design and development of the Björk-Shiley convexo-concave (BSCC) heart valve is interesting from a biomedical engineering design as well as from an ethical standpoint. In the early 1960s the earliest human prosthetic

Mechanics of Heart Valves


heart valves were developed. In 1976, Dr. Viking Björk and Shiley Incorporated developed the 60 convexo-concave BSCC-60 heart valve. The 60 designation indicates that the valve was designed to have a maximum 60 opening. The valve had improved flow characteristics over other valves that were on the market at the time. In 1979, Shiley came out with a valve that opened even farther. It was the BSCC-70 heart valve. The valve had a cobalt chrome alloy housing with a pyrolytic carbon disc, a Teflon sewing ring, and a cobalt chrome retaining strut which retained the disc in the valve.

During clinical trials of the valve, the first valve failure occurred when the outlet strut fractured. Shiley Inc. reported that failure as an anomaly and the FDA approved the valve for use in the United States. After the BSCC-60 went on the market, outlet strut fractures began to occur. Most of the reported fractures were fatal and between 1980 and 1983 Shiley implemented three voluntary recalls of the valve. During that time, Shiley’s share of the prosthetic valve market was up to 60 percent.

In the early 1980s the FDAbegan an investigation of the Björk-Shiley valve. Evidence later surfaced that Shiley Inc. attempted to delay and conceal damaging information about its product. For example, the FDA requested sets of randomly selected, recalled valves for testing. Instead of sending randomly selected valves, Shiley handpicked the valves that were sent to the FDA for testing.

Case report 1: 53-year-old German woman in 1992 in Göttingen. See Chap. 2, Sec. 2.2 for a complete description of this case report.

Case report 2: 49-year-old Japanese businessman in San Francisco in 1990. See below.

Clinical feature—case report. In 1990, a 49-year-old Japanese businessman who was attending a conference in San Francisco was brought to the emergency room of a local hospital (Chen et al., 1992). The patient was suffering from sudden onset severe difficulty breathing with anterior (front) chest pain radiating to his back. The doctors noted that he had a healed, midline sternal scar indicating previous chest surgery. The patient lacked awareness of his surroundings and he was blue around the lips. When the physicians listened to his heart and lung sounds, he had diffuse rales over both lungs and

an S3 gallop. There were no murmurs. High doses of dopamine were required to raise his systolic blood pressure to 70 mmHg.An electrocardiogram showed atrial fibrillation with a 130 beats per minute ventricular rate.

An emergency echocardiogram showed vigorous left ventricular contraction and a bulging atrium. It also showed a prosthetic heart valve but

no evidence of prosthetic disc motion. It was subsequently learned that the patient, years earlier, had been implanted with a 60°-opening BSCC prosthetic heart valve.