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A note on the critical flow to initiate closure of pivoting disc mitral valve prostheses
1. B-chama
Vol. IS. No
2. pp 151-l%.
W?I -930
1985
85 13 oa +
al
I” 198s Pcrgamon Press Ltd
Pnntcd m Great Bnum
A NOTE ON THE CRITICAL FLOW TO INITIATE CLOSURE OF PIVOTING DISC MITRAL VALVE PROSTHESES T. H. REIF and hl. C. HUFFSTL~TLERJR. KMED Institute, 3001 Laurel
Hill Ct.,
Dayton, OH 45415. U.S.A.
Abstract-Newton’s second law of motion for rotating bodies and potential flow theory is used to model the closing process of a pivoting disc prosthetic heart valve in mitral position. The model predicts closure to be dependent upon disc curvature, eccentricity. mass, diameter, density, opening angle and fluid properties. Experiments using two commercially available prostheses are shown to give good correlation with the theory for large opening angles. Divergence between theory and experiment occur at small opening angles because of the limitation of the potential flow assumption.
mathematically
NOMESCLATURE B
CL 4 ; Y H 10
buoyancy force (N) lift coefficient diameter of the occluder (m) eccentricity (m) curvature parameter (m) local acceleration of gravity (m s- *) tissue annulus diameter (m) mass moment of inertia about the
pivot
axis
(N-m-s*) lift force
(N) mass of the occluder (kg) dimensionless lifting moment volume flow rate (I. s- ‘)
L m M’
Q B r
R,
mean critical flow rate (I. s _ ‘J half-thickness of the occluder (m) radius of curvature of occluder (m) Reynolds number component of bearing reaction force in x-direction
RY
(N) component
R Re’
%
I u X Y
of bearing reaction force in y-direction
(N) standard deviation of volume Row rate data (I. s‘ ‘J time (s) mass average velocity at the valve orifice (m s- t ) distancedownstream from thecenter of the valve(m) distance from the tissue surface (m)
Greeksymbols II
opening angle (rad) opening angle required for leading edge of occluder to be tangent to Row (rad) angle of attack (rad) dimensionless eccentricity absolute viscosity of the fluid (Pa-s) mass density of the fluid (kgm-‘) mass density of the occluder (kgm“) distance along the chord from the center of the disc to the center of lift (m) dimensionless curvature
lNTRODUCI’ION
Patients receiving pecially susceptible Received 25 April
mitral valve replacements are esto cardiac arrythmias. This may be 1983; in recked form
23
October
1984.
due in part to their pre-existing cardiovascular pathology, e.g. in some institutions up to 40:1” of the patients with valvular disease develop atrial fibrillation and atrial flutter (Cordell, 1979). However, there is also an increased risk of cardiac arrythmias associated with extracorporeal circulation (Ankeney, 1979). Thus, the design of a mitral valve prosthesis should give consideration to valvular function under the adverse conditions of cardiac arrythmias. Figure I is a schematic representation of the volume flow rate in uilro through a pivoting disc mitral prosthesis, based on data from Reif and Huffstutler (1984) and Fujita et al. (I 98 I). Shaded area I depicts the volume of forward flow during diastole, area II depicts the volume of reversed flow during diastole due to the closing process of the occluder. and area III depicts the leakage between the occluder and housing after valve closure during systole. The cardiac output is the net positive area in Fig. I divided by the beat period, T,. A reduction in the principle flow phase (area I), an inefficient closing process (area II), and a prosthesis with high leakage (area III) can all reduce the cardiac output. During atrial fibrillation the atrial contractions are of high frequency and low amplitude, causing a decrease in area I. This can be a significant decrease even in the absence of valvular pathology (Hurst and Logue, 1966). The volume of reversed flow during diastole (area II) and systole (area III) now become a more significant percentage of the cardiac output and, therefore, more important design constraints. The purpose of this investigation is to develop a mathematical model which describes the closing process of pivoting disc mitral prostheses. Experimental results will be presented for the Bjork-Shiley concaveconvex (BS/CC) and the omniscience (OS) prostheses to evaluate the accuracy of the theory. These valves were chosen since they were the most geometrically similar of the commerially available pivoting disc prostheses; thus easily demonstrating the effects of disc curvature and eccentricity on closure dynamics.
T. H.
152
REIF and M.
C.
HUFFSTUTLER
E,t=S
Fig. 1. Schematic
of volume
flow rate through
pivoting
disc mitral prosthesis.
Summing the moments about the pivot axis in the clockwise direction given
MODEL
Figure 2 is a cross-sectional view of a pivoting disc prosthesis in the mitral position. The pertinent geometric parameters such as disc curvature, diameter, and eccentricity are also shown. Figure 3 is a free-body diagram of the occluder at the onset of reversed flow. The lift force, L, acts at a distance 5 from the center of the disc. The mass, m, and the buoyancy force, E, act through the center of the disc. The bearing reactions R, and R, are shown at the pivot axis. The effects of fluid friction (drag force) and the frictional moment at the pivot axis are neglected.
Lcosj({+e)-mg(l
-p/p.)
(I)
where I, is the mass moment of inertia of the occluder about the pivot axis, p the fluid density, p. the occluder density. Defining the non-dimensional lifting moment as M*
=
LcosB(S+4
(2)
!fPU2@ and the Reynolds number as Re*
u
d2B 1,~)
esin/.l=
=
y
(3)
\ \ \
FORWARD FLOW
q
TISSUE
\
REOURQITAMT
REQUROITANT
FLOW
FLOW
Fig. 2. Cross-sectional
view of prosthesis.
pivoting
disc
mitral
Fig. 3. Free-body
diagram
of occluder flow.
at onset of reversed
Pivoting
then
(lF(3)
equations
d2fi dt’=I,
may be combined
Dk pz -Re*‘M*-mg(1 [ zp
to give
1 ,
-p/p,,)ssinfi
where E is the non-dimensional
disc mitral valve prostheses
eccentricity
(4)
defined as
EL.
(5)
D,
The occluder is modeled as a two-dimensional semicircular shaped airfoil. Using potential flow theory (Streeter, 1948) the lift coefficient is C, ~ 2LID: = - 2n sin (24 - 8)
(6)
4PU
and the center of lift is
-sin
(7)
“Ok = 4 sin (24 - p)’ curvature
defined as
d =f/D,. Using equations
(8)
(5) and (6) in equation
hl’ = C,cosB(
(2) gives
+E).
(9)
Equation (4) is an ordinary, non-linear, second order, non-homogeneous differential equation with
Table
1. Design parameters
Valve
TAD(m)
BS/CC OS
0.027 0.027
0.25 0.17
‘P
r mg( I -p/p&--[P
sin/I
M
1 (10) I”
Calculation of Re* requires knowledge of 4, E. /I at the full open position, tn. po, and the fluid properties p and p. Table 1 shows this pertinent data for the size 27 mm tissue annulus diameter (TAD) BS/CC and OS prostheses. This data was used to compute C, from equation (6) and is depicted in Fig. 4 as a function of the opening angle, z. Similarly, equation (7) was used to generate Fig. 5. The results of Figs 4 and 5 were utilized to compute M* via equation (9) as shown in Fig. 6. Finally, utilizing Table 1, Fig. 6, and equation
of the BjBrk-Shiley
I# = f/D& E = e/D, 0.12 0.05
constant coefficients, the solutions of which can be obtained from most mathematical handbooks. Since the potential theory is valid only for small values of /I, pursuing the solution would be a waste of time. However, equation (4) can be used to determine the critical reversed flow required to initiate closure. If the magnitude of the reversed flow is below the critical value, then the valve will remain open fully. But if the reversed flow exceeds the critical flow, then closure is initiated. To initiate closure requires that the angular acceleration of the occluder be positive. Thus, equation (4) predicts the critical Reynolds number to be
Re* >
/I
where r#~is the non-dimensional
153
and Omniscience
m (kg) 6.6 x lo-* 6.8 x 1o-4
cardiac z_
Po(kgm-7
valve prostheses (degrees)
2.48 x IO’ 2.48 x IO’
D,(m)
59 79
0.022 0.022
3.0
-2.0
J
I
50
60
OPENING,
Fig. 4. Theoretical
80
70
a-
OEG
lift coefficient.
90
50
60 OPENING
70 ANGLE
Fig. 5. Theoretical
80 a
-DEG
center of lift
so
T. H. REIF and M C. HUFFSTUTLER
154
1.5-I
.‘a 5 : l.O-
\
37%
~=3.5x10-sPa-S
TAo=27mm
\ \ \ OS.
AT
~=l.O(lxldkg/m’
P a z : 0
-7 BLOOD
\ \
\ \
\
\
OS
50
70
60
OPENING
ANGLE
00
,Q --OEG
Fig. 6. Theoretical lifting moment.
(10) the critical Reynolds number for initiation closure was calculated. These results are shown Fig. 7 for blood and in Fig. 8 for water.
of in
ZOO-1
1 60
50
70
OPENING
ANGLE.
I I 80
I 90
a-LEG
Fig. 7. Critical Reynolds number to initiate closure with blood as working fluid.
EXPERIMENT
Two test chambers were specially machined from clear acrylic plastic to house the BS/CC and OS valves. The inside diameters of the test chambers were equal to the inside diameters of the valve housings. A special viewing port was machined and polished into the side of each chamber in order to visualize the angle of the occluder relative to the housing. The apparatus was aligned such that the body forces on the occluder were oriented in accordance with those depicted in Fig. 2.
o+o-
0
n
0
I
Ia ,
;
MAX
50
60
OPENING
ANGLE
u
MA
,X
I
I I 40
I
I
0 0
The occluder was held in the desired position from above with a thin wire to minimize the disturbance to the flow field. The angle, Q, was measured with special templates with an absolute error of f 0.02 rad. An entry length of rigid tubing with inside diameter equal to the inside diameter of the valve housing was placed upstream of the test chamber. The length of this tubing was about 30 times the inside diameter of the valve housing, ensuring a fully developed flow to the
I 70
80
90
q_DEG
Fig. 8. Critical Reynolds number to initiate closure with water as working fluid.
Pivoting
disc mitral valve prostheses
test chamber (Schlichting. 1979). A steady-state, fully developed flow of water at 2OC was delivered to the test chamber in the regurgitant flow direction as shown in Fig. 1.The volume flow rate was increased over very small increments until valve closure was initiated. The critical volume flow rate was measured with a graduated cylinder and a stop watch with an estimated absolute error of + 0.01 I. min- ‘. The experiment was run four times over the same range of opening angles for the BSCC valve. The test chamber for the OS valve was then placed in the same apparatus and the process was repeated. The resulting critical flow rates were then averaged for each opening angle. The average flow rate, Q, was then used to compute the critical Reynolds number for each opening angle via
(11)
RI?* =g.
The average percent standard deviation, (So/Q), for the BS, CC valve was 0.2 I7 and for the OS valve was 0.096. The results are summarized graphically as data points in Fig. 8.
DISCWWON
Figure 8 shows good agreement between the mathematical model and the experiment for opening angles near maximum. As the opening angle decreases the model diverges from the experimental results. This is to be expected, however, since the model was based on two-dimensional potential flow theory valid only for small angles of attack (24 - /3). Equation (10) predicts that Re* is a function of p, p, pO,m, E, M *, and a. From equation (9) we see that M* is a function of 4, E and a. Table I shows that there is little difference between the BS/CC and OS valves for m and pO. But there is considerable difference in 4, E and z,.~. Therefore, the results of Fig. 8 suggest that it is the differences in 4 and E that cause the OS valve to close at a significantly lower Re* for a = 59’. In fact, even at the full open condition, the OS valve closes at a lower Re* than does the BS/CC valve, despite the significant differences in amox. It may be of interest to note that even though both Figs 7 and 8 are non-dimensional, dynamic similitude does not seem to exist. The reason is because potential flow theory was used in the theoretical model to estimate the lifting moment, M*. Since potential flow theory is based on the assumption that the fluid is frictionless. IV* is independent of 11.The result is that the theoretical critical flow rate for closure is independent of Ii. Equation (IO) could have just as easily been written as Fl(l
The non-dimensional just by convention.
-PIP&M*
sin/?
The OS and the BS/CC prostheses utilize two different methods of retaining the occluder in the valve housing. Because of these differences, the plane of the occluder was more likely to be skewed from the pivot axis with the BSCC valve than with the OS valve. Such malalignment also skews the orientation of the hemodynamic and gravitational forces acting on the oceluder. Consequently, more variation was observed in the measurement of Q with the BS/CC prostheses. This explains why the average percent standard deviation was larger for the BS/CC valve. Figure 8 depicts relatively smooth experimental curves, probably reflecting the random skewing of the occluder and the averaging process. CONCLUSIONS
A simple model is proposed to describe the critical reversed flow rate required to initiate the closure of a pivoting disc prosthetic heart valve in mitral position. The model predicts closure to be dependent upon disc curvature, eccentricity, mass, diameter, density, opening angle and fluid properties. Experiments were conducted to evaluate the accuracy of the theory for two commercially available pivoting disc prostheses. Good correlation was observed for the full open conditions. A divergence was observed between the model and the experimental results for small opening angles because of the limitations of the theoretical assumptions. Differences in curvature and eccentricity between the Bjiirk-Shiley concave-convex and the Omniscience prostheses caused a significant difference in the minimum reversed flow required to initiate valve closure. Patients receiving mitral valve replacements are especially prone to post-operative cardiac arrythmias. During some arrythmias one can expect the principle flow phase through the valve to be significantly reduced. This reduces the cardiac output and makes the reversed flow during the closure and leakage processes a larger percentage of the already compromised cardiac output. Therefore, sustained arrythmias in patients with mitral prostheses may lead to serious ramifications because of dangerously low cardiac output if the chosen prosthesis has not been carefully designed with respect to closing dynamics and leakage. The problem of leakage has been discussed elsewhere (Reif and Huffstutler, 1980). while a first order solution of the closing dynamics problem is discussed herein. Hopefully the simple theory will serve as a stepping stone to further work in the area of more efficient design of mitral prostheses.
1 (12) ‘,I
form was used in equation
(10)
155
REFERENCES
Ankeney. J. L. (1979) Cardiac complications of extracorporeal circulation (low cardiac output syndrome). Complicarions oflntrathorocic Surgery (Edited by Cordell, A. R. and Ellison, R. G.). Little, Brown, Boston, MA. Cordell, A. R. (i979) Cardiac dysrhythmias associated with
156
T. H. REIF and
extracorporeal circulation. Complicorions of fnrrathoracic Surgery (Edited by Cordell, A. R. and Ellison, R. G.). Little, Brown, Boston, MA. Fujita, T. er al. (1981) Valve characteristics and its clinical application, especially on the biological valve. Jqx Ann. Thoroc. Surg. l(1). 30-42. Hurst, J. W. and Logue. R. B. (1966) The Heart, pp. 285-288. McGraw-Hill, New York. Reif, T. H. and Huffstutler, M. C., Jr. (1984) A comparative
M. C. HUFFSTUTLER
study of the Omniscience and the Bjiirk-Shiley cardiac valve prostheses. Int. J. Art% Organs 7, 277-282. Reif, T. H. and Huffstutler. M. C., Jr. (1980) A note on the leakage of the new Omniscience pivoting disc prosthetic heart valve. J. biomech. Engng 102, 342-344. Schlichting, H. (1979) Boundcry-Layer Theory, 7th Ed., McGraw-Hill, New York. Streeter, V. L. (1948) Fluid Dynamics. pp. 144-l 50. McGrawHill, New York.
Vol. IS. No
2. pp 151-l%.
W?I -930
1985
85 13 oa +
al
I” 198s Pcrgamon Press Ltd
Pnntcd m Great Bnum
A NOTE ON THE CRITICAL FLOW TO INITIATE CLOSURE OF PIVOTING DISC MITRAL VALVE PROSTHESES T. H. REIF and hl. C. HUFFSTL~TLERJR. KMED Institute, 3001 Laurel
Hill Ct.,
Dayton, OH 45415. U.S.A.
Abstract-Newton’s second law of motion for rotating bodies and potential flow theory is used to model the closing process of a pivoting disc prosthetic heart valve in mitral position. The model predicts closure to be dependent upon disc curvature, eccentricity. mass, diameter, density, opening angle and fluid properties. Experiments using two commercially available prostheses are shown to give good correlation with the theory for large opening angles. Divergence between theory and experiment occur at small opening angles because of the limitation of the potential flow assumption.
mathematically
NOMESCLATURE B
CL 4 ; Y H 10
buoyancy force (N) lift coefficient diameter of the occluder (m) eccentricity (m) curvature parameter (m) local acceleration of gravity (m s- *) tissue annulus diameter (m) mass moment of inertia about the
pivot
axis
(N-m-s*) lift force
(N) mass of the occluder (kg) dimensionless lifting moment volume flow rate (I. s- ‘)
L m M’
Q B r
R,
mean critical flow rate (I. s _ ‘J half-thickness of the occluder (m) radius of curvature of occluder (m) Reynolds number component of bearing reaction force in x-direction
RY
(N) component
R Re’
%
I u X Y
of bearing reaction force in y-direction
(N) standard deviation of volume Row rate data (I. s‘ ‘J time (s) mass average velocity at the valve orifice (m s- t ) distancedownstream from thecenter of the valve(m) distance from the tissue surface (m)
Greeksymbols II
opening angle (rad) opening angle required for leading edge of occluder to be tangent to Row (rad) angle of attack (rad) dimensionless eccentricity absolute viscosity of the fluid (Pa-s) mass density of the fluid (kgm-‘) mass density of the occluder (kgm“) distance along the chord from the center of the disc to the center of lift (m) dimensionless curvature
lNTRODUCI’ION
Patients receiving pecially susceptible Received 25 April
mitral valve replacements are esto cardiac arrythmias. This may be 1983; in recked form
23
October
1984.
due in part to their pre-existing cardiovascular pathology, e.g. in some institutions up to 40:1” of the patients with valvular disease develop atrial fibrillation and atrial flutter (Cordell, 1979). However, there is also an increased risk of cardiac arrythmias associated with extracorporeal circulation (Ankeney, 1979). Thus, the design of a mitral valve prosthesis should give consideration to valvular function under the adverse conditions of cardiac arrythmias. Figure I is a schematic representation of the volume flow rate in uilro through a pivoting disc mitral prosthesis, based on data from Reif and Huffstutler (1984) and Fujita et al. (I 98 I). Shaded area I depicts the volume of forward flow during diastole, area II depicts the volume of reversed flow during diastole due to the closing process of the occluder. and area III depicts the leakage between the occluder and housing after valve closure during systole. The cardiac output is the net positive area in Fig. I divided by the beat period, T,. A reduction in the principle flow phase (area I), an inefficient closing process (area II), and a prosthesis with high leakage (area III) can all reduce the cardiac output. During atrial fibrillation the atrial contractions are of high frequency and low amplitude, causing a decrease in area I. This can be a significant decrease even in the absence of valvular pathology (Hurst and Logue, 1966). The volume of reversed flow during diastole (area II) and systole (area III) now become a more significant percentage of the cardiac output and, therefore, more important design constraints. The purpose of this investigation is to develop a mathematical model which describes the closing process of pivoting disc mitral prostheses. Experimental results will be presented for the Bjork-Shiley concaveconvex (BS/CC) and the omniscience (OS) prostheses to evaluate the accuracy of the theory. These valves were chosen since they were the most geometrically similar of the commerially available pivoting disc prostheses; thus easily demonstrating the effects of disc curvature and eccentricity on closure dynamics.
T. H.
152
REIF and M.
C.
HUFFSTUTLER
E,t=S
Fig. 1. Schematic
of volume
flow rate through
pivoting
disc mitral prosthesis.
Summing the moments about the pivot axis in the clockwise direction given
MODEL
Figure 2 is a cross-sectional view of a pivoting disc prosthesis in the mitral position. The pertinent geometric parameters such as disc curvature, diameter, and eccentricity are also shown. Figure 3 is a free-body diagram of the occluder at the onset of reversed flow. The lift force, L, acts at a distance 5 from the center of the disc. The mass, m, and the buoyancy force, E, act through the center of the disc. The bearing reactions R, and R, are shown at the pivot axis. The effects of fluid friction (drag force) and the frictional moment at the pivot axis are neglected.
Lcosj({+e)-mg(l
-p/p.)
(I)
where I, is the mass moment of inertia of the occluder about the pivot axis, p the fluid density, p. the occluder density. Defining the non-dimensional lifting moment as M*
=
LcosB(S+4
(2)
!fPU2@ and the Reynolds number as Re*
u
d2B 1,~)
esin/.l=
=
y
(3)
\ \ \
FORWARD FLOW
q
TISSUE
\
REOURQITAMT
REQUROITANT
FLOW
FLOW
Fig. 2. Cross-sectional
view of prosthesis.
pivoting
disc
mitral
Fig. 3. Free-body
diagram
of occluder flow.
at onset of reversed
Pivoting
then
(lF(3)
equations
d2fi dt’=I,
may be combined
Dk pz -Re*‘M*-mg(1 [ zp
to give
1 ,
-p/p,,)ssinfi
where E is the non-dimensional
disc mitral valve prostheses
eccentricity
(4)
defined as
EL.
(5)
D,
The occluder is modeled as a two-dimensional semicircular shaped airfoil. Using potential flow theory (Streeter, 1948) the lift coefficient is C, ~ 2LID: = - 2n sin (24 - 8)
(6)
4PU
and the center of lift is
-sin
(7)
“Ok = 4 sin (24 - p)’ curvature
defined as
d =f/D,. Using equations
(8)
(5) and (6) in equation
hl’ = C,cosB(
(2) gives
+E).
(9)
Equation (4) is an ordinary, non-linear, second order, non-homogeneous differential equation with
Table
1. Design parameters
Valve
TAD(m)
BS/CC OS
0.027 0.027
0.25 0.17
‘P
r mg( I -p/p&--[P
sin/I
M
1 (10) I”
Calculation of Re* requires knowledge of 4, E. /I at the full open position, tn. po, and the fluid properties p and p. Table 1 shows this pertinent data for the size 27 mm tissue annulus diameter (TAD) BS/CC and OS prostheses. This data was used to compute C, from equation (6) and is depicted in Fig. 4 as a function of the opening angle, z. Similarly, equation (7) was used to generate Fig. 5. The results of Figs 4 and 5 were utilized to compute M* via equation (9) as shown in Fig. 6. Finally, utilizing Table 1, Fig. 6, and equation
of the BjBrk-Shiley
I# = f/D& E = e/D, 0.12 0.05
constant coefficients, the solutions of which can be obtained from most mathematical handbooks. Since the potential theory is valid only for small values of /I, pursuing the solution would be a waste of time. However, equation (4) can be used to determine the critical reversed flow required to initiate closure. If the magnitude of the reversed flow is below the critical value, then the valve will remain open fully. But if the reversed flow exceeds the critical flow, then closure is initiated. To initiate closure requires that the angular acceleration of the occluder be positive. Thus, equation (4) predicts the critical Reynolds number to be
Re* >
/I
where r#~is the non-dimensional
153
and Omniscience
m (kg) 6.6 x lo-* 6.8 x 1o-4
cardiac z_
Po(kgm-7
valve prostheses (degrees)
2.48 x IO’ 2.48 x IO’
D,(m)
59 79
0.022 0.022
3.0
-2.0
J
I
50
60
OPENING,
Fig. 4. Theoretical
80
70
a-
OEG
lift coefficient.
90
50
60 OPENING
70 ANGLE
Fig. 5. Theoretical
80 a
-DEG
center of lift
so
T. H. REIF and M C. HUFFSTUTLER
154
1.5-I
.‘a 5 : l.O-
\
37%
~=3.5x10-sPa-S
TAo=27mm
\ \ \ OS.
AT
~=l.O(lxldkg/m’
P a z : 0
-7 BLOOD
\ \
\ \
\
\
OS
50
70
60
OPENING
ANGLE
00
,Q --OEG
Fig. 6. Theoretical lifting moment.
(10) the critical Reynolds number for initiation closure was calculated. These results are shown Fig. 7 for blood and in Fig. 8 for water.
of in
ZOO-1
1 60
50
70
OPENING
ANGLE.
I I 80
I 90
a-LEG
Fig. 7. Critical Reynolds number to initiate closure with blood as working fluid.
EXPERIMENT
Two test chambers were specially machined from clear acrylic plastic to house the BS/CC and OS valves. The inside diameters of the test chambers were equal to the inside diameters of the valve housings. A special viewing port was machined and polished into the side of each chamber in order to visualize the angle of the occluder relative to the housing. The apparatus was aligned such that the body forces on the occluder were oriented in accordance with those depicted in Fig. 2.
o+o-
0
n
0
I
Ia ,
;
MAX
50
60
OPENING
ANGLE
u
MA
,X
I
I I 40
I
I
0 0
The occluder was held in the desired position from above with a thin wire to minimize the disturbance to the flow field. The angle, Q, was measured with special templates with an absolute error of f 0.02 rad. An entry length of rigid tubing with inside diameter equal to the inside diameter of the valve housing was placed upstream of the test chamber. The length of this tubing was about 30 times the inside diameter of the valve housing, ensuring a fully developed flow to the
I 70
80
90
q_DEG
Fig. 8. Critical Reynolds number to initiate closure with water as working fluid.
Pivoting
disc mitral valve prostheses
test chamber (Schlichting. 1979). A steady-state, fully developed flow of water at 2OC was delivered to the test chamber in the regurgitant flow direction as shown in Fig. 1.The volume flow rate was increased over very small increments until valve closure was initiated. The critical volume flow rate was measured with a graduated cylinder and a stop watch with an estimated absolute error of + 0.01 I. min- ‘. The experiment was run four times over the same range of opening angles for the BSCC valve. The test chamber for the OS valve was then placed in the same apparatus and the process was repeated. The resulting critical flow rates were then averaged for each opening angle. The average flow rate, Q, was then used to compute the critical Reynolds number for each opening angle via
(11)
RI?* =g.
The average percent standard deviation, (So/Q), for the BS, CC valve was 0.2 I7 and for the OS valve was 0.096. The results are summarized graphically as data points in Fig. 8.
DISCWWON
Figure 8 shows good agreement between the mathematical model and the experiment for opening angles near maximum. As the opening angle decreases the model diverges from the experimental results. This is to be expected, however, since the model was based on two-dimensional potential flow theory valid only for small angles of attack (24 - /3). Equation (10) predicts that Re* is a function of p, p, pO,m, E, M *, and a. From equation (9) we see that M* is a function of 4, E and a. Table I shows that there is little difference between the BS/CC and OS valves for m and pO. But there is considerable difference in 4, E and z,.~. Therefore, the results of Fig. 8 suggest that it is the differences in 4 and E that cause the OS valve to close at a significantly lower Re* for a = 59’. In fact, even at the full open condition, the OS valve closes at a lower Re* than does the BS/CC valve, despite the significant differences in amox. It may be of interest to note that even though both Figs 7 and 8 are non-dimensional, dynamic similitude does not seem to exist. The reason is because potential flow theory was used in the theoretical model to estimate the lifting moment, M*. Since potential flow theory is based on the assumption that the fluid is frictionless. IV* is independent of 11.The result is that the theoretical critical flow rate for closure is independent of Ii. Equation (IO) could have just as easily been written as Fl(l
The non-dimensional just by convention.
-PIP&M*
sin/?
The OS and the BS/CC prostheses utilize two different methods of retaining the occluder in the valve housing. Because of these differences, the plane of the occluder was more likely to be skewed from the pivot axis with the BSCC valve than with the OS valve. Such malalignment also skews the orientation of the hemodynamic and gravitational forces acting on the oceluder. Consequently, more variation was observed in the measurement of Q with the BS/CC prostheses. This explains why the average percent standard deviation was larger for the BS/CC valve. Figure 8 depicts relatively smooth experimental curves, probably reflecting the random skewing of the occluder and the averaging process. CONCLUSIONS
A simple model is proposed to describe the critical reversed flow rate required to initiate the closure of a pivoting disc prosthetic heart valve in mitral position. The model predicts closure to be dependent upon disc curvature, eccentricity, mass, diameter, density, opening angle and fluid properties. Experiments were conducted to evaluate the accuracy of the theory for two commercially available pivoting disc prostheses. Good correlation was observed for the full open conditions. A divergence was observed between the model and the experimental results for small opening angles because of the limitations of the theoretical assumptions. Differences in curvature and eccentricity between the Bjiirk-Shiley concave-convex and the Omniscience prostheses caused a significant difference in the minimum reversed flow required to initiate valve closure. Patients receiving mitral valve replacements are especially prone to post-operative cardiac arrythmias. During some arrythmias one can expect the principle flow phase through the valve to be significantly reduced. This reduces the cardiac output and makes the reversed flow during the closure and leakage processes a larger percentage of the already compromised cardiac output. Therefore, sustained arrythmias in patients with mitral prostheses may lead to serious ramifications because of dangerously low cardiac output if the chosen prosthesis has not been carefully designed with respect to closing dynamics and leakage. The problem of leakage has been discussed elsewhere (Reif and Huffstutler, 1980). while a first order solution of the closing dynamics problem is discussed herein. Hopefully the simple theory will serve as a stepping stone to further work in the area of more efficient design of mitral prostheses.
1 (12) ‘,I
form was used in equation
(10)
155
REFERENCES
Ankeney. J. L. (1979) Cardiac complications of extracorporeal circulation (low cardiac output syndrome). Complicarions oflntrathorocic Surgery (Edited by Cordell, A. R. and Ellison, R. G.). Little, Brown, Boston, MA. Cordell, A. R. (i979) Cardiac dysrhythmias associated with
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T. H. REIF and
extracorporeal circulation. Complicorions of fnrrathoracic Surgery (Edited by Cordell, A. R. and Ellison, R. G.). Little, Brown, Boston, MA. Fujita, T. er al. (1981) Valve characteristics and its clinical application, especially on the biological valve. Jqx Ann. Thoroc. Surg. l(1). 30-42. Hurst, J. W. and Logue. R. B. (1966) The Heart, pp. 285-288. McGraw-Hill, New York. Reif, T. H. and Huffstutler, M. C., Jr. (1984) A comparative
M. C. HUFFSTUTLER
study of the Omniscience and the Bjiirk-Shiley cardiac valve prostheses. Int. J. Art% Organs 7, 277-282. Reif, T. H. and Huffstutler. M. C., Jr. (1980) A note on the leakage of the new Omniscience pivoting disc prosthetic heart valve. J. biomech. Engng 102, 342-344. Schlichting, H. (1979) Boundcry-Layer Theory, 7th Ed., McGraw-Hill, New York. Streeter, V. L. (1948) Fluid Dynamics. pp. 144-l 50. McGrawHill, New York.