1. Biomechanics.
1977.Vol. 10.pp. 299311.
FLUID
Pclgmon
Press. Printed in Grea1 Britain
SHEAR STRESS IN PROSTHETIC VALVES* E. JOHN ROSCHKEand
EARL
C.
HEART
HARRISON
Department of Medicine, Cardiology Section, Los Angeles County. University of Southern California Medical Center, Los Angeles, CA 90033, U.S.A. AbstractA semiempirical approach has been developed to estimate the fluid shear stress developing between the valve seat and moving poppet during the opening sequence of an aortic ball valve and a disk valve prosthesis. The assumption of laminar quasisteady flow is shown to be conservative. The laminar shear stresses calculated by this method are large, and exceed threshold levels for incipient hemolysis. Results are compared to a circularorifice valve, for which much lower shear stresses are evident. Paravalvular leaks are considered as well, and shear stresses derived from a turbulent freejet analysis indicate that such leaks could lead to incipient hemolysis if significant pressure drops occur across the leak. The overall results indicate that prosthetic heart valves tend to generate a condition of mild chronic hemolysis, a condition that may not be as innocuous as has been assumed in the past.
INTRODUCTION
The degree of hemolysis that occurs in patients with normally functioning heart valve prostheses is generally mild (Walsh et al., 1969; Rodgers and Sabitson, 1969; Wallace et al., 1970; Donnelly et al., 1973) but the incidence of chronic hemolysis is rather high. For example, Crexells et al. (1972) reported on the results of 208 patients with a variety of valve types and modelsnand found that the incidence of subclinical hemolysis was 67% in that group, but severe hemolysis occurred in only 5% of the cases. Merendino and Manhas (1973) observed a marked increase in the prevalence of gallstones in a group of 39 prosthetic valve recipients, and attributed this to mild mechanical hemolysis. We have obtained similar results: the overall prevalence of gallstones in a group of 46 patients with prosthetic valves was more than three times greater than occurred in a group of patients with severe valvular heart disease. The source of gallstones associated with heart valve prostheses is thought to be red cell destruction and bilirubin production. Thus, mild mechanical hemolysis may not be as innocuous as once was supposed. There is very little quantitative data available that is useful for estimating mechanical hemolysis during flow through a heart valve prosthesis In general, most investigators have concentrated attention on the effects of turbulence produced by the poppet and valve structure. It is also reasonable, however, to consider the fluid shear stresses that develop between the poppet and the valve seat early during the opening sequence of the valve. Because of the small dimensions of this region it is unlikely that experimental velocity or shear stress will be measured in uivo, or even in uitro, for valves with timevarying flow. Blackshear (1972a. 1972b) has concluded that fluid shear * Received 21
December
1976.
stresses and turbulence are unlikely causes for mechanical hemolysis in blood flowing under all but the most severe physiological conditions. The results of theoretical calculations presented here suggest that this position may not be valid. The flow through a prosthetic valve is coupled with the poppet dynamics. Full solutions to this complex problem for timevary ing viscous flow are not yet available. The problem has been formulated for numerical computer solution by Lloyd H. Back (Jet Propulsion Laboratory, Pasadena, CA); his work has been described by Roschke (19733. Au and Greenlield (1975) have reported some fluid shear stress results for timevarying flow between the edge of a stationary diskshaped poppet and a surface representing a cage. We have used a simplified empirical approach for estimating the shear stress developing between the valve seat and a moving poppet during the opening sequence of two types of aortic prostheses: (1) ball valve and (2) disk valve. These results are compared with those obtained for (3) a circularorifice valve, i.e. a valve with a circular orifice that had an area equivalent at each instant of time to the flow area of the ball valve prosthesis. In these three cases laminar, axisymmetric flow was assumed. Paravalvular leaks have been a common source of anemia and decreased red cell survival time associated with heart valve prostheses (Herr et al., 1965; Kastor et al., 1968; Bernstein, 1971). Paravalvular leaks have been considered here as case (4), for which a turbulent freejet flow was assumed. The units used in this paper are cgs metric units. PROBLEM
FORMULATIONBALL
AND DISK VALVES
The basic approach was to assume quasisteady laminar flow at each increment of poppet displacement. Some consequences of these assumptions are discussed later. The width of the gap s between the
299
E. JOHNRO~CHKE and EARL C. HARRISON
300
poppet and its seat, and the associated flow area A,, were determined from the prescribed valve geometry as functions of the poppet displacement x. Utilizing the total poppet displacement x,, and a specified pop pet velocity u0 as boundary conditions at the end of the opening sequence t = to, prescribed functions were developed relating poppet displacement and time. The latter functions were used to determine the time history of s and A, during the opening sequence. The timevarying flow rate Q was obtained from pub lished clinical data. Thus, the time variations s(r). A,(t) and Q(c) comprised the input data. The same curve at) was used for both the ball valve and the disk valve, and for the circularorifice valve as well. Next, the input data was used to calculate the fluid velocity distribution in the gap width s for select values of time (or displacement) assuming a parabolic velocity profile at each instant of time. Coefficients for the velocity profile were determined from the continuity relation Q = judA, = i~4, where II is the local velocity normal to the flow area A, and li is the spatial average velocity. The shear stress at the wall, or at the poppet surface, was calculated from the relation r,,, = &Yu/ay), where a value of p = 0.035 poise was used for the viscosity of blood. Typical cases for an aortic ball valve and a disk valve were analyzed for valves with a primary orifice area of N 2 cm*. The model for the ball valve is shown in Fig. l(a). The flow area is the surface of the frustrum of a right circular cone with slant height s. Note that the origin of the x coordinate is fixed in space and is located at the upstream surface of the poppet when it is closed; the yaxis, however, moves with the poppet, contains the poppet center, and is normal to the valve seat. The velocity distribution is shown in Fig. l(b) where y, is the poppet radius and yz = yi + s. The velocity distribution is given by
U >> u’, h z (y, + y,)/2 and simple expressions are obtained for h, V and k in terms of Q, y, and s. The shear stress is calculated from
The maximum shear stress occurs at the surface of the valve seat and is given by 2~012

h)
(5)
k
In the case of the disk valve an attempt was made to employ the known solution for axisymmetric flow impinging on a surface normal to the direction of flow. However, this solution was discarded because it was overdetermined. Instead, the simple model shown in Fig. 2 was adopted A parabolic velocity distribution is assumed to exist at the edge of the disk such that the velocity u is parallel to the disk surface. The flow area is a cylindrical surface defined by A, = 2lrRs and the flow is Q = iiA, = 2nRsti. For this geometry the parabolic velocity profile yields
(y  h)* = k(u  V) or u = [(y  Jt)* + kV]/k,
a. MODEL FOR AORTIC Ml1 VALVE
(1)
and the boundary conditions are
Cv, h)’ = k(u'  U) (Y2  h)2=  kV,
(2)
”
(3)
where the location and magnitude of the maximum velocity are h and V respectively, k is a velocity coeficient, u’ = u cos(n/4), and u is the poppet velocity. The continuity equation is expressed as
Y’
”
Q = ii/l, = j+“2nuy cos (n/4) dy Y’I or
KY 
h)*+ Wydy.
(SEWING RING)
(4) b. PMAPOlK
The quantities h, V and k are determined by solving equations (24) and are given in Appendix A. When
VElOClTy DIS~RlUJTION IN GA?
Fig. 1. Aortic ball valve.
Fluid shear stress in prosthetic heart valves
301
valve increases. and the duration of positive pressure gradient (compared to the ejection period) is markedly increased. Poppet displacement and velocity have been measured in humans with ball valve prostheses (Gimenez er al., 1965; Winters et al., 1967) but the data is not sufficiently accurate for present purposes. AS an example. the linear displacementtime variation during the opening sequence of a mitral ball valve reported by Hamby er al. (1973) is not possible physically because it implies constant poppet velocity and zero acceleration. Alternatively, time polynomials were considered for poppet displacement and its derivatives. Thus, assume that
Fig. 2. Model for aortic disk valve with parabolic velocity distribution. a = 2U/3 so that the shear stress is the simple rela
tion IL,I = &U/(s/2) =
[email protected],
I = ci t f czt2 $ C# + c;t4 + . . .
(7)
v= g
(8)
= ci + 2c,t + 3cg2 + 4cCC3+ . .
a=d$=2c~+6c,l+
12e,t2+....
(9)
It is required that x = o = a = 0 at t = 0 and that finite values exist at t = t, (end of the opening sequence) so that if
Or IT,,,I =
3pQ/m2R.
05)
Calculations were performed for both the ball valve and the disk valve for the following poppet displacement increments: 0.1 I (x/x0) I 1.0,9 increments
then m > 2. Equation (10) is idealistic in the sense that it does not account for any poppet deformation at cage impact or for poppet rebound. If xg and v. are specified at t = ro, then it is possible to calculate two coefficients for equations (79).
0.01 I (x/x0) 5 0.1,4 increments 0 I (x/x0) 5 0.01, 7 increments.
c3 =
(4x0  v,t,)/t;
cq =  (3x,  voto)/t,4. INPUT
DATA
A basic difficulty in making flow calculations of any type for prosthetic heart valves is the lack of accurate and complete input data. In the present case simultaneous timehistories of poppet displacement and velocity and the flow through a valve of known dimensions were required. The authors were unable to find such data in the literature for humans with prosthetic valve replacement. Because poppet displacement and velocity do not necessarily scale with time, such data obtained from a patient at one heart rate cannot be coupled with flow data obtained from the same, or another, patient at a different heart rate. That is, it is perilous to attempt correcting data for heart rate. It is clear from studies in dogs (Brockman et al., 1965) that insertion of an aortic ballvalve prosthesis has a pronounced effect on the aortic pressure and flow during systole. Compared with preoperative data (normal animals), the rate of change and peak value of flow rate is reduced, the amount of aortic backflow is increased, the pressure drop across the
(11) (12)
The total displacement x0 is given by the valve geometry so that it remains to determine v0 and to. Such data has been obtained by Gibson et al. (1970) and their results are plotted for four patients in sinus rhythm in Fig. 3; lines connect points that correspond to rest and exercise states. Estimates for v. and to used here were estimated from Fig. 3. Morrow et al. (1965) measured left ventricular pressure, aortic pressure, and flow rate simultaneously in a group of patients with prosthetic aortic ball valves; the flow was measured in the ascending aorta, not at the immediate valve site. A model flow rate curve was adopted from Morrow’s data (Fig. 4). The smoothed flow curve in Fig. 4 corresponds to a patient with a StarrEdwards Model 1000 ballvalve, Size 11A, and a rest cardiac output of approx. 9 l/min at a heart rate of 86 beats per min. At that heart rate reasonable values for u0 and to are 50cm/sec and 30msec respectively for an aortic ball valve (Fig. 3). A total poppet travel of x,, = 1 cm is consistent with a size 11A ball valve. Expressions for poppet
E. JOHNROSCHKE and EARLC. HAJWSON
302
displacement and velocity were obtained for the ball valve using equations (7, 8, 11 and 12): x = 2.5 (t/L$  1.5 (r/to)4 SHADED SYMROLS OPENING TIME
01 54
I 60
I 70
0 = 250 (t/to)2  200 (t&$.
I
I
al
90
HEART RATE,
I loo
I
I 110
120
BEATS PER MINUTE
Fig. 3. Opening sequence data for aortic ball valves for four patients in sinus rhythm, after Gibson et al. (1970).
2CDl
I
I
I
0.2
Q.I
0
t
0.1
1
I
I
a.2
a3
0.4
a5
I
TIME, UC
Fig. 4. Aortic flow for a ball valve, smoothed curve for a patient with a StarrEdwards Model 1000 aortic prosthesis, adopted from Morrow et aI. (1965).
MLL
VALM
(GPNHICAL
(13) (14)
The gap width s was determined graphically from a large scale drawing of the ball valve for ten poppet positions. Corresponding values for the flow area A, were calculated using s as the slant height of the frustrum of a right circular cone; the flow area is shown in Fig. 5. The flow, or frustrum, area exceeds the primary orifice area of the valve when the poppet achieves a displacement of approx. 62% of its total excursion. The quantities s and A, were converted to time values through equation (13). Thus, all input data required for calculation of the velocity distribution across s and the wall shear stress was now available for the case of the ball valve in terms of time during the opening sequence. Although cageddisk valves are used mostly in the mitral position, it will be useful to compare ball and disk valve results for valves of the same orifice area in the aortic position. The KayShiley Series T, Size No. 3, diskvalve prosthesis has an orifice area ap proximately equivalent to the StarrEdwards Model 1000, Size llA, ballvalve prosthesis, and was used as the model here; the KayShiley disk valve is not generally used in the aortic position. This disk valve has a total poppet excursion of approx. 0.4 cm. However, poppet displacement and velocity, and flow rate data in patients does not seem to be available. From in vitro studies Cross et al. (1967) determined that disk valves achieve a higher poppet velocity and have a shorter opening time than ball valves, which is consistent with their shorter poppet excursion. It was assumed that the diskvalve values for u0 and to were 60 cm/set and 20 msec for this study. Corresponding equations for poppet displacement and velocity were calculated from equations (7, 8, 11 and 12): x = o.4(t/&J3
(15)
0 = 6O(t/t#.
(16)
For the disk valve s = x (Fig. 2) so that s and A, may be related immediately to time through equation (15) and the relation A, = ZnRs, where R is the disk radius. Flow area versus poppet position is shown in Fig. 5. For lack of other data, Fig. 4 also was used as the flow curve for the disk valve. A summary of basic input data for the ball and the disk valve is given in Table 1.
SOLUTION)
PHMMf ORIFICE
INPUT DATA AT VERY EARLY TIME DISK VA‘M @.lNuI)
0
I 0.2
I
I , I 16 0.4 DiMNSlONUSS PC+PE, PGSITION, Jr,
I D.5
,
, 1.0
0
Fig. 5. Frustrum/flow area versus poppet model ball and disk valves.
position
for
While making the present calculations it became evident that great care was necessary when prescribing input data at very early time in the opening sequence; this was generally true for 0 < x/x0 < 0.1. Values for Q and A, could not be determined accurately from Figs. 4 and 5 in that region. The following
Fluid shear stress in prosthetic heart valves
303
Table 1. Selected input values for prosthetic valves
Opening time to, msec Opening velocity u0 (at t = to), Clll/SW
Total poppet displacement x0, cm Poppet diameter 2y, and ZR, cm Primary orifice diameter D,, cm Primary orifice area &, cm2 *Name/manufacturer *Model *Size
Ball valve
Disk valve
30
20
50 1.0 1.8 1.57 1.94 StarrEdwards 1000 No. 11A
60 0.4 1.9 1.6 2.01 KayShiley Series T No. 3
in the
s = 0.395(x/x,) + 0.75(x/x,)2 A, = 1975(x/x,) + ~.S~(X/X~)~.
region
(17) (18)
A basic difficulty is that rW tends to approach infinity as t approaches zero unless the time rate of change of Q greatly exceeds the rate change of s as time approaches zero. This is evident from equations (?A, 13A) for the ball valve (Appendix), or from equation (6) for the disk valve. Essentially, z, is indeterminate at t = 0 because both Q and y and s, are zero at t = 0 but physically 7, must also be zero. This difficulty can be alleviated by forcing Q to lag the poppet motion at early time, and was accomplished by fitting a parabolic function to Q (Fig. 4) in the region 0.01 5 t 2 0.03 sec. The following equation resulted from this procedure: Q = 75,000t2 + 7250t  25,
(19)
where Q is measured in cm3/sec. the region co&esponding to Thus, in 0 < (x/x0) 5 0.1 the flow rate curve takes the following form: Ball valve, Q = 67.S&J2 Disk valve.
mm5
Fig. 6. Aortic flow curve at early time, and modified (lower) curve used for present calculations.
* Approximate commercial equivalent. functions were used for s and A, 0 < (x/x0) 5 0.1 for the ball valve:
TIM&
+ 217.5(t/t,)3  25 (20)
Q = 30(t/t,)2 + 145(t/to)3  25. (21)
Use of equations (20, 21) at early time ensures that T, will approach zero as time approaches zero. The results will be conservative, i.e. yield lower values of 7, than otherwise would be obtained. A comparison between equation (19) and the actual flow curve, which could not be determined accurately for t < 6 msec, is shown in Fig. 6. The actual flow curve is shown extrapolated to t = 0 whereas equation (19) is the lower broken curve. The existence of a timedelay before onset of flow, i.e. Q = 0 at finite time in equation (19), seems at first consideration unrealistic. physically. However, because blood has a tinite yield stress to begin with, this effect is not excluded entirely.
CIRCULARORIFICE
VALVE
Sufficient information is available to permit estimates of timevarying shear stress in natural aortic leaflet valves (Bellhouse, 1969; Swanson and Clark, 1973) using the same approach that was outlined previously for the ball and disk prosthetic valves. This information was not used, however, because a direct comparison between natural valve results and the prosthetic valve results would not have been meaningful due to differences in flow development. Instead, a hypothetical valve with a timevarying circular orifice was utilized as a control, or comparison, case. This valve resembles a natural valve only to the extent that it has a circular orifice geometry. Calculations identical to those of the prosthetic ball valve were made for the circularorifice valve; the only difference between the two sets of calculations was in the geometry of the flow area, and, therefore, the shear stress expression. The same timevarying values of A, for the ball valve were ascribed to the circularorifice valve, which had the following effective orifice diameter: D, = (4AJ/7#‘2.
(22)
For an axisymmetric pipe flow a parabolic velocity profile yields ti = U/2 so that the shear stress is calculated from the simple relafion I7.#(
=
8/U/D,.
PARAVALVULAR
(23)
LEAK
Paravalvular leaks occur adjacent to prosthetic valves when a portion of the v&lve base becomes separated or detached physically from the surrounding tissue bed due to infection, suture tear, etc. (Herr et al., 1965). A shunt flow past the valve occurs regardless of the poppet position and may give rise to a high speed, jetlike flow; when the poppet is closed (diastole for an aortic valve) a retrograde flow from the aorta will occur through the leak. For analysjs of this special case the paravalvular leak was assumed to be a short circular orifice. A turbulent jet was
E. JOHNRO~CHKE and EARLC. HARRISON
304
assumed because freejets do not remain laminar when an orifice Reynolds number Re,, = pi&,/p of approx. 30 is exceeded. It is assumed for a short orifice that a uniform velocity distribution of magnitude ti exists across the orifice exit plane. The centerline velocity U of a turbulent jet remains constant throughout an initial (potential core) region, which is followed by a transition region and then the region of developed flow. In the developed flow region the jet decays such that its centerline velocity is inversely proportional to axial distance whereas its width increases linearly with axial distance. Similarity solutions for the developed flow region have been obtained, e.g. Schlichting (1968), and arc valid for distances greater than 68 jet diameters from the source of the jet. The distribution of axial veloci’ty parallel to the jet axis is bellshaped so that a maximum shear stress in the jet occurs offaxis corresponding to the location of maximum velocity gradient. Schlichting’s results (1968) yield r,,
= CpU2,
(24)
where C = 0.01705 is dimensionless and U is the centerline velocity, which at the inception of the developed flow region is assumed to have a value of V = P. If Ap is the pressure drop across the orifice then
pii2
pu2
2
2
==
(25)
C%PP,
where C, is a discharge coefficient defined by (26)
in Fig. 7 to obtain C,,. The maximum is calculated from equations (24, 25). T,,,= = 2CC;Ap.
ReO = C,WO/P)(~AP/P)“~,
The average fluid velocity in the gap width is shown in Fig. 8 for the ball valve and the disk valve as a function of dimensionless poppe$ travel during the opening sequence. The peak value of ii is a factor of 1.8 times greater for the ball valve as compared to the disk valve. In Fig. 9 is shown the ratio of maximum fluid velocity in the gap width to the poppet velocity for both valves. The peak value of U/o attained is large for both valves and well exceeds 100; it is apparent, however, that U/o < 10 for most of the poppet travel in both cases. The wal1 shear stress developing between the poppet and its seat during the opening sequence is shown for both prosthetic valves in Fig. 10. In both cases the shear stress rises rapidly early in the opening sequence, achieves a sharp peak, and decays by several orders of magnitude in the remainder of the opening sequence. In the case of the ball valve the peak shear stress exceeds the value associated with
./‘//FOR ,,’
R.,
I
’
,
.,
”
I
, ,
,
‘I
lo3_ 4 . & 0; G . 8 2 13 
g 2
*
I)/
(27)
which may be solved for orifices of arbitrary size and pressure drop by graphical solution with the curves
(28)
RESULTS
2
C,, is a function of orifice Reynolds number and is plotted (Fig. 7) for short orifices of various length. Equation (25) may be rewritten
shear stress
1oj
,
1.1.
10z 10“ DIMNSIONLESS W??El DISPUCEMWl,
,
,
1 IO0
I/x0
Fig. 8. Average fluid velocity between poppet and seat.
z 6 SEE MILLER
AND NEMECEK
(1958)
FOR ft., >20 SEE DICKERSON AND RICE (1969)
102 10l
I
I
100
‘0’
ORIFICE
REYNOLDS
+I
I
I
102
I02
NUMRER,
R.
0
I 10’
 c”OJc
Fig, 7. Low Reynolds number discharge coefficient for shortpipe orifices with D 2 10 D,,.
0
3
DIMENSIONLESS
POPPRT DISPLACEMENT,
x/x0
Fig. 9. Ratio of maximum fluid velocity between poppet and seat to the poppet velocity.
Fluid shear stress in prosthetic heart valves
1
3
I
I I
Table 2. Summary of shear stress results for valve cases
Peak shear stress (T,J_, dyn/cm2 Timeaverage shear stress i,, dyn/cmf Ratio (GLJ?~ CIRCULAR ORIFICE
A
!
305
DlMENSlONLESS
0.4 OPENING
0.6 TIME,
t/t
0
Fig. 10. Laminar shear stress at the wall during the opening sequence.
the threshold of hemolysis (Blackshear, 1972a; Leverett et al., 1972). The peak shear stress in the disk valve was lower by more than a factor of 3 compared with the ball valve, but later in the opening sequence the disk shearstress exceeded the ball shearstress by a factor of almost 2. These differences should not be a basis for strong conclusions regarding the relative merits of the two types of prostheses because minor changes in input conditions could modify the results and even reverse them. Input conditions for the disk valve are not as reliable as for the ball valve in this study. More important is the order of magnitude of peak shear stress developed in the valves, which is high in both cases. Shear stresses towards the end of the opening sequence were small in both cases. Of particular interest is the result for the circularorifice valve in comparison with the prosthetic valves (Fig. 10). The peak shear stress in the circularorifice valve was smaller than the corresponding value for the ball valve by a factor of 48. This result is due solely to the configuration of the flow area (circular as opposed to the frustrum area of the ball valve) because the magnitude of the flow area and the flow were identical in both cases. The advantages of a central, unobstructed flow compared to an annular flow around a poppet thus are clear with respect to shear stress. It will be useful to present results in terms of an accumulated timeaverage shear stress defined by 1
‘=rT
i
s Ti
t, dT,
cm
Disk Circular valve orifice
57,390
17.830
1205
5.545
2.674
197
10.4
6.7
6.1
values of 0.1111 and 0.1665 for the ball valve and the disk valve respectively. Thus ? at any time ti < tj < to is the average shear stress over the time interval tj  ti and t0 is the timeaverage shear stress for the entire opening sequence at T = 1. Because T, ranged over orders of magnitude it was necessary to determine t by graphical integration using a minimum of 3 charts in each case. The results for the three cases are shown in Fig. 11. These curves show the same general shape as the shear stress curves in Fig 10 but they rise and decay less sharply. For the entire opening sequence the values of ?0 for the ball and disk valve exceed the value for the circularorifice valve by factors of 28, and nearly 14, respectively. A summary of shear stress results is given in Table 2 for the three cases. Shear stress results for the paravalvular leak are shown in Fig. 12 as a function of pressure drop across the orifice (leak) for orifices of various diameter. The curves are for orifices with a length to diameter ratio of unity. Except for’ very small orifices, shear stress in the turbulent jet is not a strong function of the size of the paravafvular leak. Note that the shear stresses for large Ap in Fig. 12 are of the same order of magnitude as the average shear stress over the opening sequence i0 for the ball and disk valve. The length of time that a red cell might be exposed to
10s
loz_I 0. I
where T = (t/co) is dimensionless time, AT = (t  [{)/to and ti in the present case is an initial time when TV = 0. In these calculations z = tJto had
Ball valve
I
I
I
I
I
0.2
0.4
0.6
DIMENSIONLESS
OPENING
1
I
TIME.
t/t
1.0 0
Fig. 11. Accumulated timeaverage laminar shear stress at the wall during the opening sequence.
306
E.JOHN ROSCHKE~~~EARLC.HARRISON
0 0
20
40
so
40
PESSUI DROPACROSSPAMVALWUR
l.wc w,
[email protected]
mm li#
Fig. 12. Maximum shear stress in a turbulent jet issuing from or&es of various size simulating a paravalvular leak.
these turbulent shear stresses is of interest. If it is assumed, conservatively, that the residence time is equivalent to the travel time of 8 jet diameters, then the residence time t, = 8D,/U; this assumption will be used in later discussion. This study does not delineate between the size of a leak and the accompanying pressure drop; in a given patient these parameters are related and depend on cardiac output. Large leaks tend to reduce Ap. Reynolds numbers for the various cases are defined as follows: ball valve, Re, = piu/p; disk valve, Re, = PWP = PWfl; circularorifice valve, Re, = piiD,/p = (D,/s)Re,; paravalvular leak ReO = piiD&; F = 0.035 poise and p is approximately unity for blood. For the prosthetic valves the maximum Reynolds number occurred at the end of the opening sequence; values were approx. 1360 and 720 respectively for the ball valve and the disk valve. For the circularorifice valve the maximum Reynolds number occurred early in the opening sequence and remained relatively constant thereafter at a value of approx. 4400. Reynolds number for the paravalvular leak depends on the orifice discharge coefficient and the pressure drop across the orifice; as an example, Reynolds number varied roughly as 9OOODc for Ap = 50 mm Hg and Do measured in cm.
velocity profile was used, and (4) the effects of flow acceleration were ignored. Generally, it is to be expected that larger shear stresses would develop with increased flow rates because of larger velocity gradients near the wall. It is clear from the data of Gibson et al. (1970) that larger poppet velocities and decreased opening times in patients are associated with increased cardiac output and heart rate, and posture as well (erect as compared to supine position). Because very small dimensions are involved in prosthetic valves, in terms of axial distance, it is unlikely that fully developed flow could occur between a poppet and seat even if the flow was steady and the flow rate was very large. The cases analyzed herein correspond more closely to entrance regions than to regions of fully developed flow. Lew (1973) has shown that velocity gradients in an entrance region are substantially higher compared with those that occur in the region of fully developed flow (parabolic velocity profile), and much higher shear stresses would occur near the wall in an entrance region. The real flow through a prosthetic valve involves rapid acceleration and a body (poppet) set into motion. The effects of acceleration on laminar boundary layers are well known: compared to steady flow the velocity boundary layers are thinner and have larger gradients near fixed or moving surfaces, and correspondingly higher shear stresses (Back, 1970; Back, 1971). Similar effects have been obtained for unsteady and pulsatile laminar tube flows (Daneshyar, 1970; Jones, 1972). It appears that the early effects of acceleration, impulsive motion or pulWile flow could result in shear stresses 5 to 10 times higher than a comparable steady flow. To approximate the effects of flow acceleration Swanson and Clark (1973) utilized a nonparabolic velocity distribution for their analysis of flow through a natural aortic leaflet valve: u = UC1  (r/r$l,
(30)
DISCUSSIONASSUMPTIONS AND RESULTS
where r is a radial coordinate and r. the distance from the centerline to the wall (leaflet). According to Swanson and Clark the exponent n is in the range 5 < n < 9. If equation (30) had been used in the present analysis then the simplified expression for wall shear for the ball valve (with f = 1, see Appendix), and for the disk valve, would have been
The shear stress results given previously for the prosthetic and circularorifice valves were based on quasisteady laminar, axisymmetric flow with an assumed parabolic velocity profile that is characteristic of fully developed flow. There are a number of factors indicating that the calculated shear stresses, particularly during the early portion of the valve opening sequence, are in fact too low and could be substantially higher: (1) the results were obtained for a hypothetical patient in a rest state, (2) the flow rate Q was deliberately suppressed at early time, see equation (19) and Fig. 4, (3) a fully developed laminar
This firstorder correction for flow acceleration would yield wall shear stresses n/2 times greater than those plotted in Fig. 10. There is virtually no quantitative data in the literature for shear stress in prosthetic heart valves, or data relating shear stress to levels of hemolysis, with which to compare the present theoretical results. Indeglia et al. (1%8) determined hemolysis in oitro by measuring plasma hemoglobin release for 13 different valve
Fluid shear stress in prosthetic heart valves
307
.
prostheses in both steady and pulsatile flow of fresh dog blood They did not measure shear stiesi. but found no correlation between degree of hemolysis produced, the effective orifice area of each valve, or the resistance to flow produced by each valve. The fact that their measured values for hemolysis in vitro were substantially less than occur in even mild clinical hemolysis was discussed thoroughly by Blackshear (1972a). Au and Greenfield (1975) computed shear stresses between the edge of a diskshaped poppet and a surface representing a cage. Their results are difficult to compare with our disk valve results because of differences in geometry, flow, and Reynolds number definitions. However, for a poppet velocity of 6Ocm/se~ their results indicate maximum shear stresses approaching 104dyn/cm2 in the clearance gap between the disk edge and the wall of its hypothetical cylindrical enclosure. These values are comparable to our maximum values of rW for the region between the face of the disk and its seat (Fig. 10). Some recent in vitro results for steady flow through a ball valve and tiltingdisk valve have been obtained by Yoganathan et al. (1976). lhey used a laserDoppler technique to measure velocity profiles at axial stations 2040 mm downstream of the front end of the valves and as near to the wall as 0.03 mm. Even at these relatively large axial distances from the valve seat they estimated wall shear stresses higher than lo3 dyn/cmz based on velocity profiles. Unfortunately, their technique cannot be used in the region of the valve seat itself. It is evident that numerous variables affect the mechanical hemolysis of blood near prosthetic surfaces (Blackshear, 1972a; Blackshear, 1972b) but in uiho studies suggest that hemolysis depends on the level and duration of shear stress. Incipient hemolysis i.e. the threshold at which red cell damage becomes evident, occurs at lesser levels of shear stress as the duration of application increases (Leverett et al., 1972). Richardson (1974) has attempted to predict theoretically the time for hemolysis to occur when a red cell is subjected to an arbitrary level of shear rate d (velocity gradient). His order of magnitude analysis predicts that the hemolysis time is inversely proportional to the quantity p2, or UT. The shear stress at which incipient hemolysis occurs in vitro as a function of exposure time is indicated in Fig. 13 as the shaded region, as determined by the results of Leverett et al. (1972) and other investigators. For comparison, the accumulated average shear stress t, (see equation 29) and Fig 11, is plotted also in Fig. 13 for the three valve cases analyzed herein. It is apparent that the average shear stress over the opening sequence for both the ball valve and the disk valve is comparable to the level required for incipient hemolysis, and that peak levels of t exceed that level significantly. In contrast, the circularorifice valve values are well below the region of incipient hemolysis. As indicated in Fig. 13 the region of in
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[email protected] hemolysis divides the stresstime domain into t%o regions, one in which stress e&cts dominate, and one in which surface effects dominate. Also indicated in Fig. 13 are some results for the paravalvular leak; for this case the exposure time has been taken as 8D$U. Curves have been plotted for the two values of Ap indicated, and each curve spans the range of leak sizes 0.1 5 Do 5 1 cm. These results indicate that paravalvular leaks may approach incipient hemolysis if the pressure drop across them is sufficiently severe. Nevaril et al. (1969) determined
that hemolysis occurs in uirro at shear stress levels of 3000dyn/cm2 in blood exposed for 120 set in a viscometer, but that morphological changes in red cells occur at shear stress levels as low as 1500dyn/cm2. These investigators also found that potential crushing effects on red cells that could occur during the oscillatory poppetseating of a ball valve were not significant in the hemolytic process. It is important to point out that the exposure time referred to in Fig. 13 corresponds to red cells exposed to .a single encounter. In the natural circulation red cells would be exposed to numerous encounters during their lifetime (mean lifetime of approx. 120d). According to Burton (1965) red cells could recycle through the human heart in as little as 10~ (via coronary circulation) to as long as one minute. Assuming a mean transit time of 30 set for individual red cells, a red cell would pass through the heart approx. 3.5 x 10’ times during its lifetime. The prob ability for membrane damage or lysis of individual red cells subjected to incipient or even nearincipient hemolytic shear stress levels would appear to be extremely high. Red cells that become damaged, or suffer morphological changes, also tend to suffer reduced survival time (Nevaril et al., 1969). Abnormal red cells are more easily damaged than normal red cells, and hemolysis occurs at correspondingly lower threshold shear stress levels according to Mac Calhun et al. (19753. Thus, the comparisons shown in Fig 13 are significant from an order of magnitude point
OUUTION
Of STRESS EXPOSURE,usan,+
Fig. 13. Shear stresstime plot. The region of incipient hemolysis denotes the time of duration required for the onset of hemolysis to occur at a given level of shear stress. For the three valve cases timeaverage laminar shear stress is plotted. Turbulent shear stress for two values of pressure drop are plotted for the paravalvular leak.
308
E. JOHNROSCHKE and EARL C. HARRISON
of view considering the potentially cumulative effect of numerous red cell encounters. In the case of paravalvular leaks regurgitation occurs, and the frequency of red cell encounter rises sharply when the regurgitant flow becomes a significant fraction of the forward flow (Rubinson et al., 1966). This fact coupled with the increased probability of hemolysis for damaged red cells means that the subhemolytic shear stresses ‘for paravalvular leaks (Fig. 13) cannot be dismissed without concern. Only results for the opening sequence have been obtained for the valve cases. It is not clear that comparable shear stresses would develop during the closing sequence even though, in the case of aortic prostheses, the poppet closing velocities apparently are about the same as the opening velocities. Although retrograde .flow is required for poppet closure, and some regurgitant flow past the poppet occurs before full closure, the retrograde flow velocities probably are small. If these arguments are correct, then shear stresses during closure are somewhat less than occur during the opening sequence. Mitral prostheses are more difficult to analyze because little is known quantitatively about the flow through these valves. Nolan et al. (1969) used mitral prostheses with built in flowmeters for their studies on calves. The flow acceleration in mitral valves tends to be more gentle than in aortic valves. This information coupled with the larger size of mitral valves, and correspondingly larger flow areas, suggests that shear stresses in mitral valves may not be as severe as those in aortic valves.
FURTHER DISCUSSION
Mechanical trauma to. red cells in patients with prosthetic valves frequently has been ascribed to turbulence (Kastor et al., 1968; Walsh et al., 1969; Donnelly et al., 1973) or to excessive shear stresses (Rodgers et al., 1969; Wallace et al., 1970). The hypothesis of turbulencegenerated hemolysis would appear to gain credence from the adverse effects of exercise observed in recipients of valve prostheses (Herr et al., 1965; Walsh et al., 1969; Crexells et ,al., 1972) but some investigators have expressed doubt that turbulence, by itself, is a primary cause of mechanical hemolysis (Bernstein, 1971; Crexells et al., 1972). The description of turbulence is difficult at best, and a persistent shortcoming is that a general, quantitative definition of turbulence has not yet been estab lished for periodic physiological flows. The presence of eddies or vortices in the flow is not necessarily indicative of turbulence. However, the flow patterns developed downstream of prosthetic valves, in oitro do appear to be turbulent. The color photographs, displayed in the last reference, an article describing some of the work of Swanson and Clark at Washington University in St. Louis, are among the best published. These photographs were obtained by a birefrin
gent technique, and are strong evidence in support of turbulence. The theoretical results reported herein for the short time intervals during the opening sequence of prosthetic valves suggest that very high shear stresses may develop when the flow is laminar. Indeed, because the flow is rapidly accelerating it probably remains laminar between the poppet and seat for a large portion of the opening sequence. Just downstream, however, the jetlike flow issuing from the gap width probably does experience transition to turbulent flow, especially later during systole. Many aspects of mechanical hemolysis have been discussed in detail by Blackshear (1972a, 1972b), who believes that mechanical hemolysis is due primarily to the interactions of red cells with surfaces. These interactions involve the diffusion of red cells towards surfaces, subsequent encounter, and adhesion at disturbed or active sticky sites in a process called redcell tethering. Fluid mechanic factors and wall configuration (roughness) influence the probability of red cell encounter with a wall. Red cell adhesion is affected by other factors such as protein surface layers, platelet adhesion, and clotting factors. Platelet adhesiveness has been found to increase in prosthetic valve recipients who develop significant hemolysis (Stormorken, 1971). The tethers of attached red cells may be fractured by very modest shear stresses (Blackshear et al., 1971), which may cause crenation or membrane damage and reduce lifetimes. Red cell adhesion to foreign surfaces has been studied in vim recently by Mohandas et al. (1974). These investigators measured minimum critical shear stresses for detachment of just a few dyn/cm’. Blackshear has suggested that the sites for red cell adhesion in diseased heart valves are the nonendothelialized surfaces; in the prosthetic valve, the prosthetic surface. In the case of an aortic prosthetic valve it is possible that red cells could become attached to areas of the seat or sewing ring just after valve closure and during diastole. During the early moments of valve opening these cells would be subjected to shear stresses easily capable of detaching, and perhaps damaging, them (Fig. 10). Thus, the presence of large shear stresses between the poppet and its seat may lead directly to hemolysis and would strongly influence local surfacecell interactions as well. The results given herein do not take into account the effects of wall roughness, which may modify and increase surface shear stresses compared to a smooth surface. In the case of clothcovered valves roughness elements may be present that are larger than the red cells. There is some evidence that higher levels of hemolysis may be associated with clothcovered valves as compared with smooth, bare valves (Crexells et al., 1972; Santinga et al., 1974; Wukasch et al., 1974). It is useful to estimate the size of roughness elements that are sufficient to cause disturbances in a laminar boundary, and hence might affect the shear stress. Thii is done in Appendix B. It appears that the roughness of cloth surfaces is larger than the
FIuid shear stress in prosthetic heart valves
“admissible” roughness, especially early in the opening sequence of prosthetic valves. CONCLUSIONS
Full solutions for the timedependent viscous flow through prosthetic heart valves with coupled poppet motion are not yet available. Thus, a semiempirical method was developed to estimate the order of magnitude of fluid shear stresses developing between the poppet and its seat during the opening sequence of aortic prosthetic valves. The results of this theoretical study indicate that (1) very large laminar shear stresses, probably sufficient to cause mild mechanical hemolysis, may occur between the poppet and seat of prosthetic heart valves early during the opening sequence, (2) the method of calculation is probably conservative, i.e. calculated shear stresses are too low because the effects of an accelerating flow reduce the boundarylayer thickness and would cause much steeper velocity gradients at the wall as compared to the assumed parabolic velocity profile, (3) the potential for mild mechanical hemolysis is strongly enhanced considering that iin individual red cell is exposed to numerous valve encounters during its lifetime, (4) the developing shear stress in a circular, unobstructed orifice is substantially lower than occurs in a prosthetic valve, and (5) turbulent shear stresses generated by simulated paravalvular leaks may approach incipient hemolysis levels when the pressure drop across the leak becomes sul%ciently large. The results suggest that a more sophisticated analytical approach is warranted to solve this problem more accurately, but a difficulty is that complete and accurate input data based on clinical measurements are not available.
309
Broskman, S. K., Snyder, H. E. and Collins, H. A. (1965) Dynamics of ventricular pressure. aortic pressure and flow, and their changes after insertion df the Starr aortic valve. J. Thoracic Cardiowsc. Surg. 50. 253259. Burton, A. C. (1965) Physiology and Biophysics ofthe CircuIation, p. 67. Year Bbok Gbl., Chi&g& IL. ” Crexells, C.. Aerichide. N.. Bonnv. Y.. Leoane. G. and Campeau; L. (1972) Factors iniuencing ‘ he;dolysis in valve prosthesis. Am. Heart J. 84, 161170. Cross. F. S., Akao, M. and Jones, R. D. (1967) Comparison of ball and lens heart valve prostheses. Surgery 62, 191806.
Daneshyar, H. (1970) Development of unsteady laminar flow of an incompressible fluid in a long circular pipe. Inc. J. Mech. Sci. 12, 435445. Dickerson, P. and Rice, W. (1969) An investigation of very small diameter laminar flow orifices. Tram ASME, (Series D) J. bus. Engng 91, 546548. Donnelly, R. J., Rahman, A. N., Manohitharajah, S. M., Deverall, P. B. and Watson. D. A. (1973) Chronic hemolysis following mitral valve replacementa comparison of the framemounted aortic hemogaft and the composite seat StarrEdwards prosthesis. Circulation 48, 821829.
Gibson, D. G., Broder, G. and Sowton, E. (1970) Phonocardiographic method of assffsing changes in left ventricular function after StarrEdwards replacement of aortic valve. Brit. Heart J. 32, 142148. Gimenez J. L., Winter, W. L., Davila, J. C., Connell, J. and Klein, K. S. (1965) Dynamics of the StarrEdwards ball valve prosthesis; a tinefluorographic and ultrasonic study in humans. Am. J. Med. Sci. 250, 652657. Goldstein, S. (1938) Modern Drwlopments in Fluid Mechanics, Chapter 7, Section 142. First Edition, Oxford Univ. Press, NY. Hanibv.,, R. I.. Aintablian. A. and Wisoff. B. G. (1973) Mechanism bf closure oi the mitral prosthetic valvk ani the role of atria1 systole. Am. J. Cardiol. 31. 616622. Herr, R., Starr, A., McCord, C. W. and Wood, J. A. (1965) Special problems following valve replacementembolus, leak, infection, red cell damage. Ann. Thoracic Surg. 1, 403415. Indeglia, R. A., Shea, M. A., Varco, R. L. and Bernstein, E. F. (1968) Erythrocyte destruction by prosthetic heart valves. Suppl. II to Circularion 37 and 38, 11861193. Jones, A. S. (1972) Wall shear in pulsatile flow. Bull. Math. Biophys. 34. 7986.
REFERENCES
and Greenfield. H. S. (1975) Computer graphics analysis of stresses in blood flow through a prosthetic
Au, A. D.
heart valve. Comput. Biol. Med. 4. 279291. Back, L. H. (1970) Acceleration and cooling effects in laminar boundary layerssubsonic. transonic. and supersonic speeds. AlAA J. 8, 794802. Back, L. H. (1971) A note on laminar shear flow over impulsively started bodies. Trans ASME, (Series E) J. Appl. Mech. 38. 10651068.
Bellhouse, B. J. (1969) Velocity and pressure distributions in the aortic valve. .I. Fluid Mech. 37, 587600. Bernstein, E. F. (1971) Certain aspects of blood interfacial phenomenared blood cells. Fed. Proc. 30, ISIO1515. Blackshear, P. L., Forstrom, R. J., Dorman, F. D. and Voss, G. 0. (1971) Effect of flow on cells near walls. Fed. Proc. 30. 16001609. Blackshear, P. L. (1972a) Hemolysis at prosthetic surfaces. In Chemisrry oJ Biosurfaces, Vol. 2, Chapter 11 (Edited by Hair, M. L.), pp. 523562, Marcel Dekker, NY. Blackshear. P. L. (1972b) Mechanical hemolysis in flowing blood. In BiomechanicsIts Foundations and Objectives, Chapter 19 (Edited by Fung, Y. C., Perone, N. and Anliker. M.), pp. 501528, PrenticeHall, Englewood Cliffs. NJ.
Kastor, J. A., Akabarian, M., Buckley, M. J., Dinsmore, R. E., Sanders, C. A., Scannell, J. G. and Austen, W. G. (1968) Paravalvular leaks and hemolytic anemia following insertion of StarrEdwards aortic and mitral valves. J.Thoracic
Cardiouasc. Surg. 56, 279288.
Leverett. L. B.. Hellurns. J. D.. Alfrev. C. P. and Lvnch. E. C. i1972) ‘Red blood cell damageby shear stress: Bio: php. J. 12, 257273. Lew, H. S. (1973) The use of entry flow equations in studying the uniform entry flow. J. Biomechanics 6, 205214. Mac Callum, R. N., Lynch, E. C., Hellurns, J. D. and Alfrey, C. P. (1975) Fragility of abnormal erythrocytes evaluated by response to shear stress. J. Lab. C/in. Med. 85. 6774.
Merendino, K. A. and Manhas, D. R. (1973) Manmade gallstonesa new entity following cardiac valve replacement. Ann. Surg. 177, 694703. Miller, R. P. and Nemecek, I. V. (1958) Coefficients of discharge of short pipe orifices for incompressible flow at Reynolds numbers less than one. ASME Winter Annual Meeting, Paper No. 58A106. Mohandas, N.. Hochmuth, R. M. and Spaeth. E. E. (1974) Adhesion of red cells to foreign surfaces in the presence of flow. J. Biomed. Mat. Res. 8. 119136. Morrow, A. G, Brawley, R. K. and Braunwald, E. (1965) Effects of aortic regurgitation on left ventricular perform
310
.,
E. JOHNROSCHKEand EARL C.
antedirect determinations of aortic blood flow before and after valve replacement. Suppl. _. I to Circulation 31 and 32, 180195. Nevaril. C. G.. Hellums. J. D.. Alfrev, C. P. and Lynch, C. P.’ (1969)‘Physical .effects in red. blood cell trauma: AIChE J. 15, 707711. Nolan S. P., Stewart, S., Fogarty, T. J., Dixon, S. H. and Morrow, A. G. (1969) In uivo studies of instantaneous blood flow across mitral ballvalve prostheses: effects of. cardiac output and heart rate on transvalvular energy loss. Ann. Surg. 169, 551559. Richardson, E. (1974) Deformation and haemolysis of red cells in shear flow. Proc. Roy. Sot. (Lortd.) A. 338,129153. Rodgers, B. M. and Sabitson, D. C. (1969) Hemolytic anemia following prosthetic valve replacement. Suppl. I to Circulation 39 and 40, 11151161. Roschke, E. J. (1973) An engineer’s view of prosthetic heart valve performance. Biomat., Med. Dev., Artif: Organs 1, 249290. Rubinson, R. M., Morrow, A. G. and Gebcl, P. (1966) Mechanical destruction of erythrocytes by incompetent aortic valvular prosetheses. Am Heart J. 71, 179186. Santinga, J. T., Flora, J. D., Bat&is, J. and Kirsh, M. M. (1974) Hemolysis in patients with the clothcovered aortic valve prosthesis. Am. J. Cardiol. 34, 533537. Schlichting, H. (1968) Boundary Layer Theory, pp. 699702, 6th Edition, McGrawHill, NY. Stormorken, H. (1971) Platelets, thrombosis and hemolysis Fed Proc. 30, 15511555. Swanson, W. M. and Clark, R. E. (1973) Aortic valve leaflet motion during systole: numericalgraphical determination. Circulation Res. 32. 4248. Wallace, H. W., Kenepp, D. L. and Blakemore, W.. S. (1970) Quantitation of red cell destruction associated with tivular disease and prosthetic valves. J. Thorocic Cardiovasc. Surg. 60, 842848. Walsh, f. R, Starr, A. and Ritzmann, L. W. (1969) Intravascular hemolysis in patients with prosthetic valves and valvular heart disease. Suppl. I .to Circukrrion 39 and 40, x1351140. Winters, W. L., Gimenex, 1. and Soloff, L. A. (1967) Clinical application of ultrasound in the analysis of prosthetic ball valve function. Am. J. Cardiol. 19,97107. Wukasch, D. V., Sandiford, F. M., Reul, G. J, Halhnan, G. L. and Cooley, D. A. (1974) Complications of clothcovered prosthetic valves. Am J. Cordiof. 33, 179 (abstract). Yoganathan, A. P., Corcoran, W. H. and Harrison, E. C (1976) In vitro velocity measurement in the vicinity of aortic valve prostheses. Paper presented at the 1976 AICHE Annual Meeting, Chicago. Anon. article (1974) Pretesting aortic valve prostheses Contemp. Surg. 5, 2125.
NOMENCLATURE poppet acceleration flow or frustrum area, normal to local velocity vector or&e area for paravalvular leak series coefficients for poppet displacement, equation (7) coefficient for maximum shear stress, turbulent jet, equation (24) orifice discharge coefficient, paravalvular leak, equation (26) diameter upstream of orifice, paravalvular leak effective o&ice diameter, natural valve, equation (22) diameter of orifice, paravalvular leak flow function, equation (2A), Appendix location of maximum fluid velocity, equation (1) velocity coeficient, parabolic profile, equation (1) lenath of short orifice. Fin. 7
HARRISON
velocity profile exponent, equation (30) AP pressure drop across paravalvular leak Q volume rate of flow radius of disk poppet R Re Reynolds number, see text for definitions gap width between poppet and seat. Figs. 1 and 2 S time t time at end of opening sequence (poppetcage impact) to dimensionless time, t/to dimensionless time interval (t  ti)/to, equation (29) k U fluid velocity spatial average fluid velocity P maximum fluid velocity u V poppet velocity, 0’ = ucos(n/4) x poppet displacement distance along an axis coincident with the gap width Y s and a radius vector of the ball poppet, Fig. 1 radius of ball poppet Yl sum of poppet radius and gap width, yi + s Yl dynamic viscosity, for blood, p = 0.035 poise I fluid density P a rate of shear (velocity gradient) shear stress accumulated timeaverage shear stress, equation (29) ;
n
Subscripts i corresponds to time when initial shear stress is zero max maximum value 0 denotes values when t = to (except when used with A or D) W value at the wall.
APPENDIX ABALL VALVE RELATIONS The solutions of equations (24) in the text yield the following relations for the ball valve h = fcvl + Y,)/2,
(1A)
where
( s>
fiQ  IISV’y, f=
+
2
w
< 1.
 NV’ y, + s. 3> (
JZQ
cri3fiQf nhs
1
I
(3A)
[s’(3  2f) + 6sy,(l f)]
(4A)
[
Yl(1 f) +
$2f)
[s2(3  2f) + 6sy,(l f)J
k =  %3&r kU== 
Y,(l f)+
$2 f)
[
*. I
(5A)
The velocity distribution is given by
 f) + (Y  Yl)lCS  (Y  Y,)l.
(6A)
At the surface of the valve seat the maximum shear stress is a”
6fiQf
Ir,l  P 6 = nhs I I
Yi(l I)
+ f(2 f)
I [s2(3  2f) + 6sy,(l  f)] ’
(7A)
Fluid shear stress in prosthetic heart valves For V/o’ r lO,f approaches unity so that (8A)
h = (Yr + Y2)/2 U = 3& Q/4&s
(9A)
k = nhs=/3,+/tiQ
UW
kU = (s/2)’
(11A)
u = 4V(Y  y,)Es  (Y  Y,W.
(12A)
The simplified shear stress relation becomes IT,/ = 3J?pQ/nhs’
APPENDIX BEFFECIS
= 4$/s.
(13A)
OF SURFACE ROUGHNESS
If the height E of surface protuberances project too far into a laminar boundary layer the roughness elements will produce flow disturbances in the form of vortex wakes, Roughness elements smaller than l, which is sometimes called “admissible” roughness, will not produce signi&xmt disturbances and the surface behaves as if it were hydraulically “smooth”. An old technique for making firstorder e&&es of this effect was presented by Goldstein (1938). The approach is to define and calculate a “roughness” Reynolds number R, = p&p where u, is the local velocity at the top (height E) of a roughness element: u, is obtained from a parabolic
311
velocity profile. Then, Re, < Re, where Re, is a critical Reynolds number that varies according to the shape of the roughness element. For sharpedged obstacles Re, _ 30. This method can be applied to the opening sequence of prosthetic heart valves. The flow between poppet and seat is characterized as a quasisteady channel flow at early time. The height of this channel is s, the distance between poppet and seat, which is normal to the local direction of flow. The mean velocity of the flow is li within the gap width s. Of course both P and s vary with time during the opening sequence. Using the Goldstein approach the final results for admissible roughness may be expressed in terms of the Reynolds number in the valve gap, defined as Re, = piis/p. The result is, for sharpedged roughness, (C/S)< 2.3/(Re,)*“. and E is obtained 6y calculating s and Re,, which was done to accomplish the results presented herein. Values of E were calculated throughout the opening sequence for both the ball valve and the disk valve. Shortly after opening, when s is very small, the calculated admissible roughness was only 1015 q for both valves. At full opening, z was 410 and 335~ for the ball and the disk valve respectively. The roughness of most cloth surfaces is the or&r of several hundred microns, as Iarge or larger than z. Thus, to first order, it is conceivable that clothcovered surfaces might contribute to the hemolytic process.