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Computer-assisted design of pivoting disc prosthetic mitral valves
J
THoRAc CARDIOVASC SURG
86: 126-135, 1983
Computer-assisted design of pivoting disc prosthetic mitral valves This paper describes the use of the new technique of computer testing to study the simulated performance offlat and curved pivoting disc prosthetic mitral valves. The design parameters considered are the radius of curvature of the occluder and the position of the pivot point. The performance criteria are the net stroke volume. the mean forward pressure difference. and the peak anterior velocity. The last of these criteria may be important in the prevention of small-orifice valve thrombosis. The best overall valve in the study has a radius of curvature equal to 1.5 times the diameter of the occIuder and a pivot point located 0.39 mitral-ring diameters from the anterior border of the mitral anulus. The maximum angle of opening of this optimal valve is limited to about 70 degrees by fluid-mechanical forces. Nevertheless. the inclusion of a redundant stop in the mechanical design of the valve is indicated, since the computer experiments also reveal excessive opening and failure to close in valves with nearby parameter values. These studies illustrate the usefulness of computer testing in prosthetic valve design.
David M. McQueen, Ph.D., and Charles S. Peskin, Ph.D., New York, N. Y.
Lis is the first in a series of papers on the computerassisted design of prosthetic mitral valves. In this work, we evaluated simulated prosthetic heart valves in a computer test chamber with contractile walls that model the left side of the heart. The computational method that we used to solve the coupled equations of motion of the blood, the valve, and the heart walls was described earlier.':" The physiological properties of the computer test chamber were discussed in a previous report,' which also compared the results of computer experiments and animal experiments on the natural mitral valve. For further background and additional references, the reader may want to consult Peskin's" detailed review of experimental, theoretical, and computational approaches to the fluid dynamics of heart valves. Experimental work directly related to the design of the present computational study has been performed by Kohler" and by Yoganathan and associates." Kohler directly measured the torque on the occluder of a flat pivoting disc valve held fixed at various angles in a
From Courant Institute of Mathematical Sciences. New York University. New York. N. Y. Supported by the National Institutes of Health under Research Grant H L17859. Computation also supported in part by the Department of Energy under Contract DE-AC02-76ER03077 at the Courant Mathematics and Computing Laboratory of New York. Received for publication July 2. 1982. Accepted for publication Oct. 19. 1982. Address for reprints: Charles S. Peskin. Courant Institute of Mathematical Sciences 251 Mercer Street New York. N. Y. 10012.
126
steady flow. He found that this torque goes through zero at a certain angle of opening which is less than 90 degrees. This angle will therefore be a (stable) equilibrium position for the open valve, provided that the mechanical design of the valve allows it to open that far. It is natural to speculate that the equilibrium angle, and hence the performance of the valve, will depend on the position of the pivot point. This is one of the motivations for the present study. Yoganathan and his collaborators" have used laserDoppler anemometry to measure fluid velocities in the neighborhood of tilting disc valves in vitro. With flat discs, these workers discovered a relatively stagnant region during forward flow on the side of the disc that faces the smaller opening of the valve. This region of stagnation was correlated with pathological findings on the site of valve thrombosis. These workers later showed that curvature of the disc could alleviate the problem of stagnation. For this reason, we studied the effects of curvature on stagnation in the neighborhood of the valve. In the present work, the computer is used to conduct parametric studies on flat and curved pivoting disc valves in the mitral position. The parameters studied are the position of the pivot point and the radius of curvature of the occluder. For each combination of pivot point and curvature, the computer predicts the performance of the valve, and the predicted performance is evaluated in terms of net stroke volume, mean forward pressure difference, and peak anterior velocity (which we believe is related to the problem of stagnation
Volume 86 Number 1 July. 1983
and small-orifice valve thrombosis as discussed earlier). This work was inspired by the widespread clinical use of tilting disc valves."! However, we did not attempt to model any particular commercial valve. Certain details of the design of these valves are beyond the scope of the computational method and other details have been modified or omitted to focus attention on the two parameters of the present study. For example, the computational mesh used in these studies does not have sufficient resolution to distinguish between a sharp-edged occluder and an occluder with a rounded edge. Moreover, the model valves are pivoted at a definite point; the occluder does not slide forward as in some of the tilting disc valves. Similarly, we did not impose any constraint on the maximum angle of opening in our model valves. This omission makes it possible to study the influence of the fluid dynamics on the maximum angle of opening. In summary, our philosophy is to model a generic type of prosthetic valve and to study certain aspects of the design problem for this class of valves. There are also certain general limitations of the computational method which the reader should keep in mind when using the results of the present study. Briefly, these are that the model is two-dimensional and that the computations are performed at a Reynolds number which has been arbitrarily reduced by a factor of 25. Nevertheless, the results are realistic despite these limitations.' We do not doubt that these limitations will have some influence on the quantitative aspect of the results. For example, the computed pressure differences appear to be lower than experimental pressure differences; we attribute this effect to the two-dimensional character of the model. Lowering the Reynolds number increases the relative importance of fluid viscosity, and this has a smoothing effect on the flow. Nevertheless, we believe that the method can be used to make valid comparisons between different valves that have been computer-tested under identical conditions, especially when these valves are all of the same general type, as in the present study. Since the model is two-dimensional, the pivot axis of the valve is necessarily perpendicular to the plane of the model left heart. This plane has been chosen so that it bisects the two major leaflets of the natural mitral valve and so that it includes the apex of the left ventricle as well as the ventricular outflow tract (Fig. 1).3 Thus we are not free to rotate the valve within the mitral anulus; the pivot axis must be parallel to the base of the anterior leaflet of the natural mitral valve. Two choices still remain: The pivot can be placed on the anterior or on the posterior side of the valve. We have chosen the first of
Computer-assisted design of prosthetic valves
I27
Fig. 1. Design parameters of pivoting disc valves: Diameter of the mitral anulus (dm). diameter of the occluder (do), radius of curvature of the occluder (r], and position of the pivot point (x p) . The dimensionless parameters of this study are R = ro/d o and X, = xp/d m •
these alternatives. Thus, in this study, the motion of the occluder resembles that of the anterior leaflet of the natural mitral valve, and flow through the larger opening of the valve is directed somewhat toward the posterior wall of the left ventricle. Another restriction on the scope of the present study is that all of the valves are evaluated under a particular set of conditions, which we regard as physiological resting conditions for the canine left heart. (The choice of the canine heart facilitates comparison with experimental results, since heart valve experiments are often performed in dogs.) In practice, it is important to know how heart valves will function under exercise conditions. Therefore, a natural extension of the present work will be a study in which the parameters of the model left heart are varied to simulate exercise. In the present work, however, we use a single set of physiological conditions and the variables are the design parameters of the valve. Methods A computer is used to solve the equations of blood motion in the heart. The heart is treated as though it were immersed in fluid, and the assumption is made that the blood, valves, heart muscle, and external fluid all have the same constant density. This makes it possible to
128
The Journal of Thoracic and Cardiovascular Surgery
McQueen and Peskin
formulate the mathematical problem in such a way that the valves and heart muscle appear as special regions of space where extra forces are applied to an otherwise homogeneous fluid. These forces give the valves and the heart walls their material and physiological properties. The fluid velocity at any given time is stored on a computational mesh with 64 points in each direction. The configuration of the immersed boundary (valves and heart muscle) is stored in terms of the coordinates of about 300 boundary points. A typical time step of the computational method proceeds as follows. At the beginning of each time step, the fluid velocity at each mesh point and the boundary configuration are known. Boundary forces are computed and applied to the fluid. The fluid velocity is changed under the influence of the boundary forces. Then the boundary is moved at the new fluid velocity. This completes the time step. Each computational experiment uses 640 time steps to span the part of the cardiac cycle that includes ventricular diastole, atrial systole, and early ventricular systole. This is the time period of greatest importance for the fluid dynamics of the mitral valve. Details of the computational method 1,2 and of the physiological aspects of the model left heart' have been given elsewhere. We now turn to a discussion of the design parameters of the model valve. These parameters are defined in Fig. 1, which shows the valve in its closed position in the model left heart at the beginning of the computations (end-systole) . In Fig. 1, the line AB represents the plane of the mitral anulus. The valve is constructed as a circular arc CPD whose center 0 lies on the atrial side of the line AB and whose endpoints C,D are symmetrically placed on this line. The valve is pivoted about the point P, and P' is the projection of the pivot point onto the plane of the mitral ring. Let: d; = Diameter of the mitral ring (AB). do = Diameter of the occluder (CD). r, = Radius of curvature of the occluder. xp = Distance along the plane of the mitral ring from its anterior border to the projection of the pivot point (AP'). Then the geometry of the valve is completely determined (except for scale) when we specify the dimensionless parameters:
R = fo/do X, = xp/d m G = Y2(d m - do}/dm For example: R = 00 means that the occluder is flat, whereas R = 0.5 means that it takes the form of a semicircle. Similarly, X p = G means that the occluder is pivoted at the anterior end of the valve, whereas
X, = 0.5 means that the pivot point is in the center of the occluder. In this study, we consider only pivot points on the anterior half of the occluder; thus G « x, <0.5. The parameter G determines the relative size of the gaps at the ends of the valve. Throughout this study, G = 0.06. These gaps each correspond to one meshwidth on the computational mesh that is used to solve the equations of fluid dynamics. Since each boundary point in the valve and heart wall has an effective radius of about one mesh-width, these gaps are effectively closed when the occluder is in the closed position. The model valve is equipped with a stop which prevents it from rotating past the closed position during ventricular systole. However, there is no corresponding stop to limit the maximum angle of opening during ventricular diastole. As a result, some of the model valves become caught in the open position and fail to close. Such valves will be called "incompetent" since they exhibit massive regurgitation leading to a very small net stroke volume. The significance of this mode of failure for the design of pivoting disc valves will be discussed at the end of the Results and discussion section. The parameters Rand X, are the crucial design parameters of the present study. Our goal is to determine the best choice of these parameters. To do this, we shall examine four indices of predicted valve performance as functions of Rand Xs, These are the net stroke volume, the peak opening angle, the mean forward pressure difference, and the peak anterior velocity. The net stroke volume is the net volume of fluid that crosses the plane of the mitral anulus while the mitral valve is open. This is the difference between the forward volume and the regurgitant volume associated with closure of the valve. If the valve fails to close, we stop integrating the flow at the end of the computational experiment, but the result is a very low net stroke volume. Such results define the incompetent valves in our study. The peak opening angle is the largest angle of opening achieved by the valve during the computational experiment. We do not use this angle to choose between different valves, since the peak opening angle has no significance in itself. We present results on peak opening angle because it helps to explain the dependence of the other performance indices on the design Rarameters. The mean forward pressure difference (Llp) is defined as T Llp =
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Volume 86 Number 1 July. 1983
Computer-assisted design of prosthetic valves 1 2 9
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Fig. 3. Peak opening angle (8*) as a function of pivot point position at various curvatures. At a given curvature, the valve opens further when it is pivoted further from the center of the occluder. At a given pivot point, increased curvature (decreased R) increases the peak opening angle.
where t = 0 is the time when flow begins at the onset of diastole and t = T is the first zero-crossing of the flow, which occurs during early ventricular systole. These two times are chosen to eliminate the inertial contribution to mean forward pressure difference.' The peak anterior velocity i~ essentially the maximum velocity that occurs on the anterior- side of the occluder. The precise definition of peak anterior velocity is as follows. First, let Qa(t) be the volume rate of flow through the anterior opening of the valve and let Aa be the area of this opening. Then define
explains the interaction between fluid dynamics and blood clotting, it is impossible to say exactly how stagnation should be measured, but it seems reasonable to assume that low peak velocities are the essence of "stagnation." Thus we try to design the valve in such a way that the peak anterior velocity is as large as possible. In summary, the computational results will be presented in terms of plots of net stroke volume, peak opening angle, mean forward pressure difference, and peak anterior velocity. These quantities will be plotted against position of the pivot point (Xs) for selected values of the radius of curvature (R). The incompetent valves in our study are distinguished by exceptionally low net stroke volumes. Among the competent valves, we seek the values of Rand X, that maximize peak anterior velocity while minimizing mean forward pressure difference. Since computer testing is a new technique in prosthet-
v.(t) = Q.(t)/A.
Thus va(t) is the mean velocity on the anterior side of the valve at time t. Finally, let peak anterior velocity be the maximum over t of va(t). The significance of peak anterior velocity is its relation to the problem of stagnation and small-orifice valve thrombosis. In the absence of a theory that
1 30
The Journal of Thoracic and Cardiovascular Surgery
McQueen and Peskin
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ic valve design, it may be appropriate to comment on its CJSt. Each of the valve tests reported in this paper required 40 minutes of central processor time on a CDC 6600. The cost of such a computational experiment is $160. Since we report On 17 different valves in the present paper, One could say that the cost of the study was $2,720.
Results and discussion Fig. 2 shows the computed values of net stroke volume as a function of the position of the pivot point (Xs) for various radii of curvature (R = 00, 1.5, 1.0, and 0.75.) The valves that we have tested fall into two well-separated groups. The competent valves all have a net stroke volume on the order of 20 em', and the incompetent valves have a net stroke volume that is ten times smaller. The phenomenon of incompetence will be discussed in detail later; briefly, it involves massive regurgitation that arises because the occluder opens too far and becomes caught in the open position. All of the flat valves that we have tested are competent. The
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curved valves all have a critical pivot point at which a rapid transition from competence to incompetence occurs. The sensitivity of this transition is very striking in Fig. 2: The net stroke volume changes by a factor of 10, and the change in the position of the pivot point is about 3% of the diameter of the mitral ring. Notice, too, that the place where this transition occurs varies in a systematic way with the curvature of the valve. Greater curvatures (smaller R) require that the valve be pivoted closer to its center for competent performance. We have also tested semicircular valves (R = 0.5, not shown in the figure.) These are incompetent at all positions of the pivot point. Since the incompetent valves have to be rejected, they are marked x in Fig. 2 and in all of the figures that follow. The valves that remain are each given a unique identifying letter (a to j) for ease of identification from one figure to the next.
Volume 86 Number 1
Computer-assisted design of prosthetic valves 1 3 1
July . 1983
Fig. 6. Computed streamlines of the best competent valve at each curvature .
Fig. 3 summarizes the effect of the design parameters on the motion of the valve. At each curvature, the peak opening angle is greater when the valve is pivoted closer to the anterior border of the niitral ring. At a given pivot point, increased curvature (decreased R) leads to increased values of the peak opening angle. All of the competent valves but one have peak opening angles that are less than 90 degrees. (The exception, valve f, has a peak opening angle ~ 100 degrees). The incompetent valves all have much larger peak opening angles, which approach or exceed 180 degrees. Fig. 4 shows that the mean forward pressure drop can be reduced by placing the pivot point closer to the anterior border of the mitral ring (i.e., further from the center of the valve). For the curved valves, however, this criterion conflicts with the overriding criterion of com-
petent closure. If the incompetent valves marked x are rejected, then Fig. 4 may be used to identify the competent valve with the lowest value of mean forward pressure drop at each curvature. These are valves a, for g, i, and j. Among these we notice that valves a, f, and g all have a mean forward pressure drop es 2 mm Hg, while valves i and j have substantially larger values of mean forward pressure drop (3.5 and 5 mm Hg, respectively). (These values of mean forward pressure drop would be greatly increased under exercise conditions.) Fig. 5 contains the most interesting results of the study. It shows the relationship between pivot point, curvature, and peak anterior velocity. (As explained earlier, we regard low peak anterior velocity as a sign that the fluid is stagnant on the anterior side of the
The Journal of Thoracic and Cardiovascular Surgery
132 McQueen and Peskin
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occluder, and we seek high values of peak anterior velocity in an effort to prevent small-orifice valve thrombosis.) At each curvature, the peak anterior velocity has a clear maximum at a certain pivot point. Moreover, the optimal pivot point shifts toward the center of the valve as the curvature is increased (decreasing R), and the value of peak anterior velocity at the optimal pivot point increases with increasing curvature. If peak anterior
velocity were the only design criterion, one would conclude that the curvature should be very large and that the pivot point should be close to the center of the valve. In fact, however, the incompetent valves (marked x) must be rejected. When this is done, the highest values of peak anterior velocity achieved by the competent valves are about 48 cm/ sec. Valvesf. g, i, and j all perform similarly in this respect. Weare now in a position to choose the best overall
Volume 86 Number 1
Computer-assisted design of prosthetic valves
July, 1983
133
Fig. 8. Computed streamlines of the best overall valve in this study.
valve. Valves f and g are competent, and they achieve very nearly the lowest values of mean forward pressure drop and the highest values of peak anterior velocity of any of the competent valves in this study. None of the other valves satisfies all of these criteria simultaneously. We can choose between f and g by recalling that g has a stroke volume that is about 10% higher and a mean forward pressure difference that is about 10% lower than f Thus valve g (R = 1.5, X, = 0.39) is the best overall valve in this study. At this point it seems appropriate to remind the reader of the principal limitations of the present study. The model is two-dimensional, the Reynolds number is artificially low, the model valves do not correspond in every detail to clinical valves in current use, and the valves are tested under a fixed set of (resting) conditions. Given these limitations, we believe that experimental confirmation of our findings should be obtained before the results of this study are applied in practice. The optimal design found here can be used as a starting point for an experimental study aimed at improving the design of pivoting disc valves. To what extent does the curvature of valve g contribute to its superior performance? To answer this question, we should compare valve g with valve d. which is the flat valve pivoted at the same value of Xs, The net stroke volume is very similar for these two valves. The mean forward pressure difference, however, is 35%
lower for valve g and the peak anterior velocity is 40% higher. The latter figure is similar to the 60% improvement in minor-orifice velocity reported by Yoganathan and associates," who compared flat-disc and curved-disc Bjork-Shiley valves by laser-Doppler anemometry. Yoganathan's group did not find any difference in the pressure drop of the two Bjork-Shiley valves, however. (Complete agreement with Yoganathan's results should not be expected because of significant differences in the valves themselves and in the conditions of the tests.) Although Figs 2 to 5 are useful for comparison of different valves, they fail to show detailed information that is available from our computational experiments. Figs. 6 and 7 present the computed streamlines and waveforms (velocity, pressure, phonocardiogram, and angle of opening as functions of time) of four selected valves: c. g. i, and j. These are the competent valves with the highest peak anterior velocity at each curvature. Fig. 6 shows the computed streamlines of the four selected valves at 75 msec intervals during ventricular diastole and early ventricular systole. The density of the streamlines is proportional to the speed of flow, and the direction of the streamlines indicates the direction of flow. Note the prominent vortices that appear in the ventricle. In the case of the flat valve (c) only a single vortex appears. This vortex (which rotates clockwise in Fig. 6) is shed from the downstream margin of the
134
McQueen and Peskin
The Journal of Thoracic and Cardiovascular Surgery
Fig. 9. Computed streamlines of an incompetent valve. Note that the curvature is the same as in Fig. 8 and that the pivot point has been moved anteriorly by only 6% of the diameter of the mitral ring.
occluder. The curved valves (g. i, and j) each shed an addition vortex from the anterior border of the mitral ring. This secondary vortex also rotates clockwise, and it merges with the primary vortex later in diastole. Fig. 7 shows computed waveforms corresponding to the streamlines of Fig. 6. From top to bottom. these waveforms are the mitral velocity (which is defined as the mitral flow divided by the area of the prosthetic mitral anulus), the left atrial and ventricular pressures, a computed phoncardiogram, and a plot of the opening angle as a function of time. Again , it is clear that valve g has the largest overall forward flow, the smallest overall pressure difference, and the largest overall angle of opening. (Valve g has about the same peak opening angle as valve c. but it stays near its peak opening angle for a longer period of time.) An unfavorable aspect of valve g is the greater backflow associated with closure. From the stroke volume data (Fig. 2) we see that this is more than compensated by the forward flow. Valve g has the largest net stroke volume in the study . A trend that should be noticed in Fig. 7 is the increasing delay in the onset of forward flow and valve opening as the curvature is increased (decreasing R) and as the valve is pivoted closer to its center. Another interesting point is the prominent third heart sound that appears in all of the phonocardiographic records during the period of rapid filling.
Fig. 8 shows the computed streamlines of the best overall valve, g. at 25 msec intervals during ventricular diastole and early ventricular systole. Although the peak opening angle is only about 70 degrees, the curvature of the valve has the effect that the direction of the upstream end of the occluder is nearly perpendicular to the valve ring when the valve is maximally open (frame 3). This may explain the substantial anterior flow that is evident in frames 3 to 5. We now tum to a more detailed discussion of tne phenomenon of incompetence, or failure to close. In our study all of the flat pivoting discs are competent , but some of the curved pivoting discs appear to open too far and become caught in the open position at the onset of ventricular systole. An example (R = 1.5, X, = 0.33) is shown in Fig. 9. This valve has the same curvature as valvesf and g. which are the two best valves in the study; it differs from valve f by a shift of pivot point equal to 3% of the diameter of the mitral ring. Clearly, excessive opening of the valve could be prevented by including a stop that limits the angle of opening. All of the valves in clinical use have such a stop, and we believe that such a stop is indeed necessary for safety even if the limitation imposed by the stop is not actually reached during the normal operation of the valve. The inclusion of a stop in the mechanical design of the valve introduces a third design parameter, maximum
Volume 86 Number 1
Computer-assisted design ofprosthetic valves 1 3 5
July, 1983
opening angle, that we have not considered in the present study. It appears, though, that the inclusion of this parameter will not lead to better overall performance than that of the best valves considered here. Our reasoning is as follows. Curvature increases the peak opening angle of the valve (see Fig. 3). For any curved valve with an opening angle that is actually limited by a stop (that is, its opening angle in the absence of the stop would be greater than the maximum angle allowed by the stop), there must be another valve with a smaller curvature and the same pivot point that willachieve the same angle of opening. Our guess is that the latter valve would give better performance by the criteria of this study. It would also "wear" better since the impact on the stop would be smaller or even zero. These ideas lead us to suggest the design concept of a redundant stop in which the opening of the valve is normally limited by fluid dynamic forces but the mechanical design of the valve ensures that further opening does not occur in unusual circumstances.
Conclusions The best overall valve in this study has an occluder with a radius of curvature equal to 1.5 times its diameter and a pivot point located 0.39 mitral-ring diameters from the anterior border of the mitral anulus. Although this optimal valve has a fluid-mechanical restriction in its angle of opening to about 70 degrees (under the conditions of the present study), we think it is important to include a redundant stop in the mechanical design of the valve itself to prevent further opening and possible catastrophic failure to close. The importance of this is underscored by our finding of excessive opening and massive regurgitation for valves with nearby parameter values. Further computational work is needed to check whether the optimal design determined here is still
optimal under exercise conditions, and further experimental confirmation of our findings should be obtained before our results are applied in practice. We hope this computational study will encourage further experimentation. We are indebted to Edward L. Yellin, R. W. M. Frater, Alexandre J. Chorin, and Olof B. Widlund for their substantive contributions to this work during the years in which the methods were developed. We also would like to thank Terry C. Moore for providing a superb computing environment at the Courant Institute.
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REFERENCES Peskin CS: Numerical analysis of blood flow in the heart. J Comput Phys 25:220-252, 1977 Peskin CS, McQueen DM: Modeling prosthetic heart valves for numerical analysis of blood flow in the heart. J Comput Phys 37:113-132, 1980 McQueen DM, Peskin CS, Yellin EL: Fluid dynamics of the mitral valve. Physiological aspects of a mathematical model. Am J Physiol 242:HI095-HI I 10, 1982 Peskin CS: The fluid dynamics of heart valves. Experimental, theoretical, and computational methods. Ann Rev Fluid Mech 14:235-259, 1982 Kohler J: Opening angle and torque ofthe Bjork-Shiley and Lillehei-Kaster heart valve prostheses. Proceedings of the European Society for Artificial Organs, Vol 2, W. Berlin, 1975, Westkreuz, pp 33-35 Yoganathan AP, Reamer HH, Corcoran WH, Harrison EC: The Bjork-Shiley aortic prosthesis. Flow characteristics of the present model vs. the convexo-concave model. Scand J Cardiovasc Surg 14:1-5, 1980 Bjork VO, Henze A: Prosthetic heart valve replacement. Nine years' experience with the Bjork-Shiley tilting disc valve, Tissue Heart Valves, MI Ionescu, ed., Boston, 1979, Butterworth & Co., Ltd., pp 1-28 Lillehei CW, Kaster RL, Bloch JH: Clinical experience with the new central flow pivoting disc aortic and mitral prosthesis. Chest 60:298, 1971
THoRAc CARDIOVASC SURG
86: 126-135, 1983
Computer-assisted design of pivoting disc prosthetic mitral valves This paper describes the use of the new technique of computer testing to study the simulated performance offlat and curved pivoting disc prosthetic mitral valves. The design parameters considered are the radius of curvature of the occluder and the position of the pivot point. The performance criteria are the net stroke volume. the mean forward pressure difference. and the peak anterior velocity. The last of these criteria may be important in the prevention of small-orifice valve thrombosis. The best overall valve in the study has a radius of curvature equal to 1.5 times the diameter of the occIuder and a pivot point located 0.39 mitral-ring diameters from the anterior border of the mitral anulus. The maximum angle of opening of this optimal valve is limited to about 70 degrees by fluid-mechanical forces. Nevertheless. the inclusion of a redundant stop in the mechanical design of the valve is indicated, since the computer experiments also reveal excessive opening and failure to close in valves with nearby parameter values. These studies illustrate the usefulness of computer testing in prosthetic valve design.
David M. McQueen, Ph.D., and Charles S. Peskin, Ph.D., New York, N. Y.
Lis is the first in a series of papers on the computerassisted design of prosthetic mitral valves. In this work, we evaluated simulated prosthetic heart valves in a computer test chamber with contractile walls that model the left side of the heart. The computational method that we used to solve the coupled equations of motion of the blood, the valve, and the heart walls was described earlier.':" The physiological properties of the computer test chamber were discussed in a previous report,' which also compared the results of computer experiments and animal experiments on the natural mitral valve. For further background and additional references, the reader may want to consult Peskin's" detailed review of experimental, theoretical, and computational approaches to the fluid dynamics of heart valves. Experimental work directly related to the design of the present computational study has been performed by Kohler" and by Yoganathan and associates." Kohler directly measured the torque on the occluder of a flat pivoting disc valve held fixed at various angles in a
From Courant Institute of Mathematical Sciences. New York University. New York. N. Y. Supported by the National Institutes of Health under Research Grant H L17859. Computation also supported in part by the Department of Energy under Contract DE-AC02-76ER03077 at the Courant Mathematics and Computing Laboratory of New York. Received for publication July 2. 1982. Accepted for publication Oct. 19. 1982. Address for reprints: Charles S. Peskin. Courant Institute of Mathematical Sciences 251 Mercer Street New York. N. Y. 10012.
126
steady flow. He found that this torque goes through zero at a certain angle of opening which is less than 90 degrees. This angle will therefore be a (stable) equilibrium position for the open valve, provided that the mechanical design of the valve allows it to open that far. It is natural to speculate that the equilibrium angle, and hence the performance of the valve, will depend on the position of the pivot point. This is one of the motivations for the present study. Yoganathan and his collaborators" have used laserDoppler anemometry to measure fluid velocities in the neighborhood of tilting disc valves in vitro. With flat discs, these workers discovered a relatively stagnant region during forward flow on the side of the disc that faces the smaller opening of the valve. This region of stagnation was correlated with pathological findings on the site of valve thrombosis. These workers later showed that curvature of the disc could alleviate the problem of stagnation. For this reason, we studied the effects of curvature on stagnation in the neighborhood of the valve. In the present work, the computer is used to conduct parametric studies on flat and curved pivoting disc valves in the mitral position. The parameters studied are the position of the pivot point and the radius of curvature of the occluder. For each combination of pivot point and curvature, the computer predicts the performance of the valve, and the predicted performance is evaluated in terms of net stroke volume, mean forward pressure difference, and peak anterior velocity (which we believe is related to the problem of stagnation
Volume 86 Number 1 July. 1983
and small-orifice valve thrombosis as discussed earlier). This work was inspired by the widespread clinical use of tilting disc valves."! However, we did not attempt to model any particular commercial valve. Certain details of the design of these valves are beyond the scope of the computational method and other details have been modified or omitted to focus attention on the two parameters of the present study. For example, the computational mesh used in these studies does not have sufficient resolution to distinguish between a sharp-edged occluder and an occluder with a rounded edge. Moreover, the model valves are pivoted at a definite point; the occluder does not slide forward as in some of the tilting disc valves. Similarly, we did not impose any constraint on the maximum angle of opening in our model valves. This omission makes it possible to study the influence of the fluid dynamics on the maximum angle of opening. In summary, our philosophy is to model a generic type of prosthetic valve and to study certain aspects of the design problem for this class of valves. There are also certain general limitations of the computational method which the reader should keep in mind when using the results of the present study. Briefly, these are that the model is two-dimensional and that the computations are performed at a Reynolds number which has been arbitrarily reduced by a factor of 25. Nevertheless, the results are realistic despite these limitations.' We do not doubt that these limitations will have some influence on the quantitative aspect of the results. For example, the computed pressure differences appear to be lower than experimental pressure differences; we attribute this effect to the two-dimensional character of the model. Lowering the Reynolds number increases the relative importance of fluid viscosity, and this has a smoothing effect on the flow. Nevertheless, we believe that the method can be used to make valid comparisons between different valves that have been computer-tested under identical conditions, especially when these valves are all of the same general type, as in the present study. Since the model is two-dimensional, the pivot axis of the valve is necessarily perpendicular to the plane of the model left heart. This plane has been chosen so that it bisects the two major leaflets of the natural mitral valve and so that it includes the apex of the left ventricle as well as the ventricular outflow tract (Fig. 1).3 Thus we are not free to rotate the valve within the mitral anulus; the pivot axis must be parallel to the base of the anterior leaflet of the natural mitral valve. Two choices still remain: The pivot can be placed on the anterior or on the posterior side of the valve. We have chosen the first of
Computer-assisted design of prosthetic valves
I27
Fig. 1. Design parameters of pivoting disc valves: Diameter of the mitral anulus (dm). diameter of the occluder (do), radius of curvature of the occluder (r], and position of the pivot point (x p) . The dimensionless parameters of this study are R = ro/d o and X, = xp/d m •
these alternatives. Thus, in this study, the motion of the occluder resembles that of the anterior leaflet of the natural mitral valve, and flow through the larger opening of the valve is directed somewhat toward the posterior wall of the left ventricle. Another restriction on the scope of the present study is that all of the valves are evaluated under a particular set of conditions, which we regard as physiological resting conditions for the canine left heart. (The choice of the canine heart facilitates comparison with experimental results, since heart valve experiments are often performed in dogs.) In practice, it is important to know how heart valves will function under exercise conditions. Therefore, a natural extension of the present work will be a study in which the parameters of the model left heart are varied to simulate exercise. In the present work, however, we use a single set of physiological conditions and the variables are the design parameters of the valve. Methods A computer is used to solve the equations of blood motion in the heart. The heart is treated as though it were immersed in fluid, and the assumption is made that the blood, valves, heart muscle, and external fluid all have the same constant density. This makes it possible to
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McQueen and Peskin
formulate the mathematical problem in such a way that the valves and heart muscle appear as special regions of space where extra forces are applied to an otherwise homogeneous fluid. These forces give the valves and the heart walls their material and physiological properties. The fluid velocity at any given time is stored on a computational mesh with 64 points in each direction. The configuration of the immersed boundary (valves and heart muscle) is stored in terms of the coordinates of about 300 boundary points. A typical time step of the computational method proceeds as follows. At the beginning of each time step, the fluid velocity at each mesh point and the boundary configuration are known. Boundary forces are computed and applied to the fluid. The fluid velocity is changed under the influence of the boundary forces. Then the boundary is moved at the new fluid velocity. This completes the time step. Each computational experiment uses 640 time steps to span the part of the cardiac cycle that includes ventricular diastole, atrial systole, and early ventricular systole. This is the time period of greatest importance for the fluid dynamics of the mitral valve. Details of the computational method 1,2 and of the physiological aspects of the model left heart' have been given elsewhere. We now turn to a discussion of the design parameters of the model valve. These parameters are defined in Fig. 1, which shows the valve in its closed position in the model left heart at the beginning of the computations (end-systole) . In Fig. 1, the line AB represents the plane of the mitral anulus. The valve is constructed as a circular arc CPD whose center 0 lies on the atrial side of the line AB and whose endpoints C,D are symmetrically placed on this line. The valve is pivoted about the point P, and P' is the projection of the pivot point onto the plane of the mitral ring. Let: d; = Diameter of the mitral ring (AB). do = Diameter of the occluder (CD). r, = Radius of curvature of the occluder. xp = Distance along the plane of the mitral ring from its anterior border to the projection of the pivot point (AP'). Then the geometry of the valve is completely determined (except for scale) when we specify the dimensionless parameters:
R = fo/do X, = xp/d m G = Y2(d m - do}/dm For example: R = 00 means that the occluder is flat, whereas R = 0.5 means that it takes the form of a semicircle. Similarly, X p = G means that the occluder is pivoted at the anterior end of the valve, whereas
X, = 0.5 means that the pivot point is in the center of the occluder. In this study, we consider only pivot points on the anterior half of the occluder; thus G « x, <0.5. The parameter G determines the relative size of the gaps at the ends of the valve. Throughout this study, G = 0.06. These gaps each correspond to one meshwidth on the computational mesh that is used to solve the equations of fluid dynamics. Since each boundary point in the valve and heart wall has an effective radius of about one mesh-width, these gaps are effectively closed when the occluder is in the closed position. The model valve is equipped with a stop which prevents it from rotating past the closed position during ventricular systole. However, there is no corresponding stop to limit the maximum angle of opening during ventricular diastole. As a result, some of the model valves become caught in the open position and fail to close. Such valves will be called "incompetent" since they exhibit massive regurgitation leading to a very small net stroke volume. The significance of this mode of failure for the design of pivoting disc valves will be discussed at the end of the Results and discussion section. The parameters Rand X, are the crucial design parameters of the present study. Our goal is to determine the best choice of these parameters. To do this, we shall examine four indices of predicted valve performance as functions of Rand Xs, These are the net stroke volume, the peak opening angle, the mean forward pressure difference, and the peak anterior velocity. The net stroke volume is the net volume of fluid that crosses the plane of the mitral anulus while the mitral valve is open. This is the difference between the forward volume and the regurgitant volume associated with closure of the valve. If the valve fails to close, we stop integrating the flow at the end of the computational experiment, but the result is a very low net stroke volume. Such results define the incompetent valves in our study. The peak opening angle is the largest angle of opening achieved by the valve during the computational experiment. We do not use this angle to choose between different valves, since the peak opening angle has no significance in itself. We present results on peak opening angle because it helps to explain the dependence of the other performance indices on the design Rarameters. The mean forward pressure difference (Llp) is defined as T Llp =
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Volume 86 Number 1 July. 1983
Computer-assisted design of prosthetic valves 1 2 9
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Fig. 3. Peak opening angle (8*) as a function of pivot point position at various curvatures. At a given curvature, the valve opens further when it is pivoted further from the center of the occluder. At a given pivot point, increased curvature (decreased R) increases the peak opening angle.
where t = 0 is the time when flow begins at the onset of diastole and t = T is the first zero-crossing of the flow, which occurs during early ventricular systole. These two times are chosen to eliminate the inertial contribution to mean forward pressure difference.' The peak anterior velocity i~ essentially the maximum velocity that occurs on the anterior- side of the occluder. The precise definition of peak anterior velocity is as follows. First, let Qa(t) be the volume rate of flow through the anterior opening of the valve and let Aa be the area of this opening. Then define
explains the interaction between fluid dynamics and blood clotting, it is impossible to say exactly how stagnation should be measured, but it seems reasonable to assume that low peak velocities are the essence of "stagnation." Thus we try to design the valve in such a way that the peak anterior velocity is as large as possible. In summary, the computational results will be presented in terms of plots of net stroke volume, peak opening angle, mean forward pressure difference, and peak anterior velocity. These quantities will be plotted against position of the pivot point (Xs) for selected values of the radius of curvature (R). The incompetent valves in our study are distinguished by exceptionally low net stroke volumes. Among the competent valves, we seek the values of Rand X, that maximize peak anterior velocity while minimizing mean forward pressure difference. Since computer testing is a new technique in prosthet-
v.(t) = Q.(t)/A.
Thus va(t) is the mean velocity on the anterior side of the valve at time t. Finally, let peak anterior velocity be the maximum over t of va(t). The significance of peak anterior velocity is its relation to the problem of stagnation and small-orifice valve thrombosis. In the absence of a theory that
1 30
The Journal of Thoracic and Cardiovascular Surgery
McQueen and Peskin
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ic valve design, it may be appropriate to comment on its CJSt. Each of the valve tests reported in this paper required 40 minutes of central processor time on a CDC 6600. The cost of such a computational experiment is $160. Since we report On 17 different valves in the present paper, One could say that the cost of the study was $2,720.
Results and discussion Fig. 2 shows the computed values of net stroke volume as a function of the position of the pivot point (Xs) for various radii of curvature (R = 00, 1.5, 1.0, and 0.75.) The valves that we have tested fall into two well-separated groups. The competent valves all have a net stroke volume on the order of 20 em', and the incompetent valves have a net stroke volume that is ten times smaller. The phenomenon of incompetence will be discussed in detail later; briefly, it involves massive regurgitation that arises because the occluder opens too far and becomes caught in the open position. All of the flat valves that we have tested are competent. The
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curved valves all have a critical pivot point at which a rapid transition from competence to incompetence occurs. The sensitivity of this transition is very striking in Fig. 2: The net stroke volume changes by a factor of 10, and the change in the position of the pivot point is about 3% of the diameter of the mitral ring. Notice, too, that the place where this transition occurs varies in a systematic way with the curvature of the valve. Greater curvatures (smaller R) require that the valve be pivoted closer to its center for competent performance. We have also tested semicircular valves (R = 0.5, not shown in the figure.) These are incompetent at all positions of the pivot point. Since the incompetent valves have to be rejected, they are marked x in Fig. 2 and in all of the figures that follow. The valves that remain are each given a unique identifying letter (a to j) for ease of identification from one figure to the next.
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Computer-assisted design of prosthetic valves 1 3 1
July . 1983
Fig. 6. Computed streamlines of the best competent valve at each curvature .
Fig. 3 summarizes the effect of the design parameters on the motion of the valve. At each curvature, the peak opening angle is greater when the valve is pivoted closer to the anterior border of the niitral ring. At a given pivot point, increased curvature (decreased R) leads to increased values of the peak opening angle. All of the competent valves but one have peak opening angles that are less than 90 degrees. (The exception, valve f, has a peak opening angle ~ 100 degrees). The incompetent valves all have much larger peak opening angles, which approach or exceed 180 degrees. Fig. 4 shows that the mean forward pressure drop can be reduced by placing the pivot point closer to the anterior border of the mitral ring (i.e., further from the center of the valve). For the curved valves, however, this criterion conflicts with the overriding criterion of com-
petent closure. If the incompetent valves marked x are rejected, then Fig. 4 may be used to identify the competent valve with the lowest value of mean forward pressure drop at each curvature. These are valves a, for g, i, and j. Among these we notice that valves a, f, and g all have a mean forward pressure drop es 2 mm Hg, while valves i and j have substantially larger values of mean forward pressure drop (3.5 and 5 mm Hg, respectively). (These values of mean forward pressure drop would be greatly increased under exercise conditions.) Fig. 5 contains the most interesting results of the study. It shows the relationship between pivot point, curvature, and peak anterior velocity. (As explained earlier, we regard low peak anterior velocity as a sign that the fluid is stagnant on the anterior side of the
The Journal of Thoracic and Cardiovascular Surgery
132 McQueen and Peskin
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occluder, and we seek high values of peak anterior velocity in an effort to prevent small-orifice valve thrombosis.) At each curvature, the peak anterior velocity has a clear maximum at a certain pivot point. Moreover, the optimal pivot point shifts toward the center of the valve as the curvature is increased (decreasing R), and the value of peak anterior velocity at the optimal pivot point increases with increasing curvature. If peak anterior
velocity were the only design criterion, one would conclude that the curvature should be very large and that the pivot point should be close to the center of the valve. In fact, however, the incompetent valves (marked x) must be rejected. When this is done, the highest values of peak anterior velocity achieved by the competent valves are about 48 cm/ sec. Valvesf. g, i, and j all perform similarly in this respect. Weare now in a position to choose the best overall
Volume 86 Number 1
Computer-assisted design of prosthetic valves
July, 1983
133
Fig. 8. Computed streamlines of the best overall valve in this study.
valve. Valves f and g are competent, and they achieve very nearly the lowest values of mean forward pressure drop and the highest values of peak anterior velocity of any of the competent valves in this study. None of the other valves satisfies all of these criteria simultaneously. We can choose between f and g by recalling that g has a stroke volume that is about 10% higher and a mean forward pressure difference that is about 10% lower than f Thus valve g (R = 1.5, X, = 0.39) is the best overall valve in this study. At this point it seems appropriate to remind the reader of the principal limitations of the present study. The model is two-dimensional, the Reynolds number is artificially low, the model valves do not correspond in every detail to clinical valves in current use, and the valves are tested under a fixed set of (resting) conditions. Given these limitations, we believe that experimental confirmation of our findings should be obtained before the results of this study are applied in practice. The optimal design found here can be used as a starting point for an experimental study aimed at improving the design of pivoting disc valves. To what extent does the curvature of valve g contribute to its superior performance? To answer this question, we should compare valve g with valve d. which is the flat valve pivoted at the same value of Xs, The net stroke volume is very similar for these two valves. The mean forward pressure difference, however, is 35%
lower for valve g and the peak anterior velocity is 40% higher. The latter figure is similar to the 60% improvement in minor-orifice velocity reported by Yoganathan and associates," who compared flat-disc and curved-disc Bjork-Shiley valves by laser-Doppler anemometry. Yoganathan's group did not find any difference in the pressure drop of the two Bjork-Shiley valves, however. (Complete agreement with Yoganathan's results should not be expected because of significant differences in the valves themselves and in the conditions of the tests.) Although Figs 2 to 5 are useful for comparison of different valves, they fail to show detailed information that is available from our computational experiments. Figs. 6 and 7 present the computed streamlines and waveforms (velocity, pressure, phonocardiogram, and angle of opening as functions of time) of four selected valves: c. g. i, and j. These are the competent valves with the highest peak anterior velocity at each curvature. Fig. 6 shows the computed streamlines of the four selected valves at 75 msec intervals during ventricular diastole and early ventricular systole. The density of the streamlines is proportional to the speed of flow, and the direction of the streamlines indicates the direction of flow. Note the prominent vortices that appear in the ventricle. In the case of the flat valve (c) only a single vortex appears. This vortex (which rotates clockwise in Fig. 6) is shed from the downstream margin of the
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The Journal of Thoracic and Cardiovascular Surgery
Fig. 9. Computed streamlines of an incompetent valve. Note that the curvature is the same as in Fig. 8 and that the pivot point has been moved anteriorly by only 6% of the diameter of the mitral ring.
occluder. The curved valves (g. i, and j) each shed an addition vortex from the anterior border of the mitral ring. This secondary vortex also rotates clockwise, and it merges with the primary vortex later in diastole. Fig. 7 shows computed waveforms corresponding to the streamlines of Fig. 6. From top to bottom. these waveforms are the mitral velocity (which is defined as the mitral flow divided by the area of the prosthetic mitral anulus), the left atrial and ventricular pressures, a computed phoncardiogram, and a plot of the opening angle as a function of time. Again , it is clear that valve g has the largest overall forward flow, the smallest overall pressure difference, and the largest overall angle of opening. (Valve g has about the same peak opening angle as valve c. but it stays near its peak opening angle for a longer period of time.) An unfavorable aspect of valve g is the greater backflow associated with closure. From the stroke volume data (Fig. 2) we see that this is more than compensated by the forward flow. Valve g has the largest net stroke volume in the study . A trend that should be noticed in Fig. 7 is the increasing delay in the onset of forward flow and valve opening as the curvature is increased (decreasing R) and as the valve is pivoted closer to its center. Another interesting point is the prominent third heart sound that appears in all of the phonocardiographic records during the period of rapid filling.
Fig. 8 shows the computed streamlines of the best overall valve, g. at 25 msec intervals during ventricular diastole and early ventricular systole. Although the peak opening angle is only about 70 degrees, the curvature of the valve has the effect that the direction of the upstream end of the occluder is nearly perpendicular to the valve ring when the valve is maximally open (frame 3). This may explain the substantial anterior flow that is evident in frames 3 to 5. We now tum to a more detailed discussion of tne phenomenon of incompetence, or failure to close. In our study all of the flat pivoting discs are competent , but some of the curved pivoting discs appear to open too far and become caught in the open position at the onset of ventricular systole. An example (R = 1.5, X, = 0.33) is shown in Fig. 9. This valve has the same curvature as valvesf and g. which are the two best valves in the study; it differs from valve f by a shift of pivot point equal to 3% of the diameter of the mitral ring. Clearly, excessive opening of the valve could be prevented by including a stop that limits the angle of opening. All of the valves in clinical use have such a stop, and we believe that such a stop is indeed necessary for safety even if the limitation imposed by the stop is not actually reached during the normal operation of the valve. The inclusion of a stop in the mechanical design of the valve introduces a third design parameter, maximum
Volume 86 Number 1
Computer-assisted design ofprosthetic valves 1 3 5
July, 1983
opening angle, that we have not considered in the present study. It appears, though, that the inclusion of this parameter will not lead to better overall performance than that of the best valves considered here. Our reasoning is as follows. Curvature increases the peak opening angle of the valve (see Fig. 3). For any curved valve with an opening angle that is actually limited by a stop (that is, its opening angle in the absence of the stop would be greater than the maximum angle allowed by the stop), there must be another valve with a smaller curvature and the same pivot point that willachieve the same angle of opening. Our guess is that the latter valve would give better performance by the criteria of this study. It would also "wear" better since the impact on the stop would be smaller or even zero. These ideas lead us to suggest the design concept of a redundant stop in which the opening of the valve is normally limited by fluid dynamic forces but the mechanical design of the valve ensures that further opening does not occur in unusual circumstances.
Conclusions The best overall valve in this study has an occluder with a radius of curvature equal to 1.5 times its diameter and a pivot point located 0.39 mitral-ring diameters from the anterior border of the mitral anulus. Although this optimal valve has a fluid-mechanical restriction in its angle of opening to about 70 degrees (under the conditions of the present study), we think it is important to include a redundant stop in the mechanical design of the valve itself to prevent further opening and possible catastrophic failure to close. The importance of this is underscored by our finding of excessive opening and massive regurgitation for valves with nearby parameter values. Further computational work is needed to check whether the optimal design determined here is still
optimal under exercise conditions, and further experimental confirmation of our findings should be obtained before our results are applied in practice. We hope this computational study will encourage further experimentation. We are indebted to Edward L. Yellin, R. W. M. Frater, Alexandre J. Chorin, and Olof B. Widlund for their substantive contributions to this work during the years in which the methods were developed. We also would like to thank Terry C. Moore for providing a superb computing environment at the Courant Institute.
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REFERENCES Peskin CS: Numerical analysis of blood flow in the heart. J Comput Phys 25:220-252, 1977 Peskin CS, McQueen DM: Modeling prosthetic heart valves for numerical analysis of blood flow in the heart. J Comput Phys 37:113-132, 1980 McQueen DM, Peskin CS, Yellin EL: Fluid dynamics of the mitral valve. Physiological aspects of a mathematical model. Am J Physiol 242:HI095-HI I 10, 1982 Peskin CS: The fluid dynamics of heart valves. Experimental, theoretical, and computational methods. Ann Rev Fluid Mech 14:235-259, 1982 Kohler J: Opening angle and torque ofthe Bjork-Shiley and Lillehei-Kaster heart valve prostheses. Proceedings of the European Society for Artificial Organs, Vol 2, W. Berlin, 1975, Westkreuz, pp 33-35 Yoganathan AP, Reamer HH, Corcoran WH, Harrison EC: The Bjork-Shiley aortic prosthesis. Flow characteristics of the present model vs. the convexo-concave model. Scand J Cardiovasc Surg 14:1-5, 1980 Bjork VO, Henze A: Prosthetic heart valve replacement. Nine years' experience with the Bjork-Shiley tilting disc valve, Tissue Heart Valves, MI Ionescu, ed., Boston, 1979, Butterworth & Co., Ltd., pp 1-28 Lillehei CW, Kaster RL, Bloch JH: Clinical experience with the new central flow pivoting disc aortic and mitral prosthesis. Chest 60:298, 1971