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Methods of design of leaflet valvular prostheses
Methods of design of leaflet valvular prostheses 1 Stresses in the mitral valve leaflets in health and disease Richard E. Clark, M.D., and Salvatore P. Sutera, Ph.D., St. Louis, Mo.
Wth
the gradual but steady improvement in biomaterials, attention has once again turned to exploring the possibilities of a leaflet type of prosthetic valve. The desirability of a central-flow device which avoids the turbulence caused by central occluding mechanisms has long been understood. However, the major concern with leaflet prostheses has been the durability of such devices. Previous laboratory and clinical trials used leaflet prosthetic valves designed on an emperic basis. Long durability of a flexing device can only be achieved by knowledge of the stresses within the normal valve leaflet. No such analysis has appeared to date although experimental stress-strain data for the valves of both man and dog have been published. One method currently under investigation is the use of close-range stereophotogrammetry to obtain the surface character in global coordinates, from which the stresses may be obtained by complex computer processing. Although extremely accurate, the method is time consuming and expensive. As an alternative, classic matheFrom the Division of Cardiothoracic Surgery, Department of Surgery, and the Department of Mechanical and Aerospace Engineering, Washington University, St. Louis, Mo. 63110. Supported in part by U. S. Public Health Service Grants HL 13803-02 and HE 12839-03. Received for publication March 12, 1973.
890
matic techniques were applied to the leaflets of the mitral valve and matched to existing experimental data. This is a report of the analysis. Methods Assumptions. A wide spectrum exists with respect to size and shape of the normal human mitral valve. Therefore, a general analysis was used to accommodate this spectrum based on plausible assumptions. The calculated stresses within any leaflet will depend on the morphologic details and actual dimensions. The first assumption was (I) that the shape of the annulus during peak systole was that of an indented ellipse similar to that of the kidney and (2) that the primary curvature of the leaflets was in a direction across the valve (Fig, I), i.e., transverse to the line of leaflet coaptation. This assumption required that the curvature of the plane of the long axis of the annulus be negligible. This simplication could be justified only if the junctional zones of tissue at the commissures were ignored and the analysis was confined to an idealized infinitesimal strip in the mid-zone of the valve. Unless the commissural zones have significantly greater curvature, the three-dimensional character should not account for an increase of an
Volume 65
Design of leaflet valvular prostheses
Num ber 6
89 1
June. 1973
Left Atrium
/
Annulus
Aortic
I
Mural
CHORDAE TENDINEAE FORCES
=====:;:t dl Leaflet Leaflet
t
Ventricle Fig. 2. Idealized cross-sectional view of the human mitral valve at strip dl. The components of the tension exerted by the chordae tendineae are assumed to be normal to the plane of the annulus.
•
a
----....,.~I
1 I
I I
I I I
I
Fig. 1. View of a normal human mitral valve from the atrial side. The analysis was confined to an idealized infinitesimal strip (dt) in the mid-zone of the valve.
order of magnitude in stresses in those zones. The second assumption concerned the state of stress in the leaflets. If the leaflets are very thin and compliant, and if the minimum radius of curvature under load is much greater than the thickness of the leaflets, than shear stresses and bending moments in the leaflets can be neglected. Correspondingly, the bending moments in the leaflets were assumed to be negligible. These are the classical assumptions in the analysis of stress in very thin shells or membranes. As a consequence , the state of internal stress is one of pure tension which is uniform across the thickness of the membrane. The third assumption was (1) that the forces exerted by the chordae tendineae can be represented as a distributed tension along the free edges of the leaflets and (2) that the component of this tension on the crosssectional plane (Fig. 2) was approximately normal to the plane of the annulus. The annular attachment of the leaflets was visualized as a mechanically ideal hinge, i.e., a
Pressure Loading p. Py - Po
CadI y Fig. 3. Free -body diagram of strip dl. P. , Ventricular pressure . P., Atrial pressure. For other abbreviations, see Appendix.
hinge which offers no resistance to rotation. Finally, the instantaneous force balance at the peak of systole was treated as quasistatic, and inertial forces were neglected . This was justified on the grounds of small leaflet mass and /or negligible acceleration at the instant of peak ventricular systole. Stress analysis. The mathematics of the stress analysis may be found in the Appendix. A free-body diagram (Fig. 3) of a cross section from an infinitesimally thin strip of each leaflet was constructed. Equal lengths of leaflet coaptation were assumed . Assuming a static equilibrium on an arbitrary differential element of the membrane (Fig. 4) at given left ventricular and left
The Journal of
892
Clark and Sutera
Thoracic and Cardiovascular Surgery
dll ..
00
= 1.42 em. - - - - - I
So = 2.0em
P = 130 mm Hg
y Fig. 4. Representation of static equilibrium on an arbitrary differential element of the membrane.
1 - - - - a., 1 . 3 7 e m . -
.
am' 0.72 em.
Fig. 6. The geometry of the human mitral valve in acute left ventricular dilatation.
aortic
Sm· I . O em
So = 2.0 em P
= 130 mm Hg
p=
130 mm Hg
Fig. 5. The geometry of the idealized normal human mitral valve.
Fig. 7. The geometry of a ballooning mitral valve in the floppy valve syndrome.
atrial pressures, it was possible to calculate the internal forces (force per unit length) by developing a series of ordinary differential equations. These first led to the conclusion that the magnitude of the tension within the leaflet was constant along the entire strip taken. It is important to note that the leaflets were assumed to have a curved shape with the valve fully closed. Existing anatomic data for leaflet lengths and coaptation lengths were used. The multiple equations were solved by trial-and-error analysis with the use of a computer. Two additional cases were then considered by means of the same techniques. These were (I) acute left ventricular dilatation and (2) the "floppy" or ballooning mitral valve. In the first case, the primary change was assumed to be in
the long axis of the left ventricle; hence, the leaflets were pulled down into the ventricle, significantly altering the shape of the mitral valve at closure. In the second condition, the floppy valve, only changes in the lengths of the transverse leaflet and of the chordae tendineae were altered. A leaflet thickness of 0.5 mm. was assumed in all cases. These differences are compared in Figs. 5 to 7. The results of the analysis are summarized in Table I. Since the tension was found to be equal throughout the strip analyzed, the total chordal force was calculated by multiplying by an assumed leaflet length. The papillary muscle forces were calculated by adding one half of each total leaflet force. These computations allowed comparison to published experimental data."
Volume 65
Design of leaflet valvular prostheses
Number 6
893
June, 1973
Table I. Stresses in human mitral valves
Valve condition
Normal Acute left ventricular dilatation Floppy valve syndrome
Aortic leaflet Mural leaflet (in 10 5 dynes/ (in 10 5 dynes/ sq. cm.) sq. em.)
24.6 33.2 21.0
Discussion Continued efforts toward the development of a central-flow device for the mitral position seem justified because of functional hydrodynamic deficiencies of prostheses." To this end, appropriate guidelines must be developed for both the valvular design and the choice of materials for a central-flow device. Therefore, an analysis of the lineal tensions, leaflet stresses, and total forces encountered by the chordae tendineae is required for the normal human mitral valve. The data derived from the present analysis were compatible with previous stress-strain studies," which indicated that the valve operates in a range of anisotropic and then isotropic behavior. Furthermore, the data of Wieting and colleagues, G derived from an experimental study of normal human valves, showed that the summed forces for the papillary muscles in normal valves were 7.60 x 10" dynes." The value derived from our studies is 5.7 x 10" dynes for the normal component of total chordal tension, calculated from assumed values of valvular dimensions from the literature. Small adjustments of the assumed leaflet edge lengths and/or a traction-angle factor could result in exact matching of the experimental data. The two additional cases considered are interesting because of the clinical analogies and because they provide a frame of reference for the normal case. Further, the marked difference in stress found for the condition of acute ventricular dilatation demonstrates that the geometry of the valve leaflet is highly important with respect to the stresses sustained.
Normal component of chordal tension, each leaflet (10 5 dynes)
13.6 36.0 9.6
Aortic
I
Mural
Normal component of total chordal tension (in 10 5 dynes)
3.7
2.0
5.7
5.0
5.4 1.4
10.4
3.2
4.6
In left ventricular dilatation, the summed forces carried by the chordae tendineae were nearly twice those of the normal valve (93 per cent increase). It is unlikely that the chordae tendineae would rupture, even with marked distortion of the mitral valvular apparatus in this condition. However, in the presence of previous disease (subacute or acute bacterial endocarditis) and/or concurrent disease of the collagen matrix of the chordae, a significant increase in tensile stress could conceivably cause rupture. Also, the possibility of a simultaneous increase in the traction angle, as proposed by Burch and Giles, I would aggravate the resultant stress levels in the individual chordae. The magnification factor is a cosine function and results in increased values for chordal loads compared to the load calculated in this report.
Summary Well-known mathematic techniques, comprising membrane theory and computer analysis, have been used to calculate the internal stresses at peak ventricular systole in the aortic and mural leaflets of the normal mitral valve and in two cases of altered geometry-acute left ventricular dilatation and the floppy mitral valve. The results demonstrate that, for the assumptions made and the dimensions used, the calculated peak stresses correspond closely to those found experimentally for the normal valve. Further, calculations for simulated pathologic clinical conditions showed that the stresses in the leaflets are nearly doubled in the case of acute left ven-
894
The Journal of Thoracic and Cardiovascular Surgery
Clark and Sutera
tricular dilatation but are reduced in the case of the floppy mitral valve. This study has aided the design and selection of suitable materials for use in prosthetic leaflet heart valves.
and dT d8 -tanfl+T--p ds ds
2 3
4
5
6
dT (cot fl + tan 8) = O. ds This can only be satisfied for any 8 if dT
-
C:.
it.
ments and at the free edge be and These vectors are directed along the local tangents to the membrane curve. 2. The profile of the membrane curve will be described in terms as a function of 8(s) where s is the curvilinear coordinate (Fig. 3). 3. Taking an element length (Fig. 4) ds, located at point Q, forces T(s) and T(s + ds) act on the element and represent the internal tensions exerted by neighboring elements. 4. Static equilibrium of the element requires the sum of the instantaneous forces to be zero. Therefore ~
F,
=
0,
~
r,
=
= O.
(4)
constant.
(5)
ds
Therefore
T Therefore d8
P
ds
T
(6)
Integration of Eq (6) yields
e
(s)
Ps
:-= -
T
+ 8
0
)
7r
fl = - at s 2
S.
Thus 8
0
7r
P
2
T
=- -
S.
P(S - s).
2
dT cot ds
e -
d8 T - + P ds
o
(2)
(9)
T
Since dy ds
dx = Sin 8, - = Cos fl ds
(10)
Integration yields the x and y coordinates x (s) =JCos 8 ds, y (s) =JSin 8 ds. (11)
Substitution of Eq (9) into (11) yields
and
(8)
Eq (7) now becomes
(1)
0
(7)
0 •
This angle (8 was eliminated in favor of the total strip length S, measured along the strip from the annulus to the coaptation point by inserting the condition
Appendix Formulation. I. Let the lineal tensions at the annular attach-
(3)
adding Eq (2) and (3)
REFERENCES Rusted, I. E., Scherfley, C. H., and Edwards, J. E.: Studies of the Mitral Valve. I. Anatomic Features of the Normal Mitral Valve and Associated Structures, Circulation 6: 825, 1952. Chiechi, M. A., Lees, W. M., and Thompson, R.: Functional Anatomy of the Normal Mitral Valve, 1. THoRAe. SURG. 32: 378, 1956. Nolan, S. P., Stewart, S., Fogarty, T. J., Dixon, S. H., Jr., and Morrow, A. G.: In Vivo Studies of Instantaneous Blood Flow Across Mitral Ball-Valve Prostheses: Effects of Cardiac Output and Heart Rate on Transvalvular Energy Loss, Ann. Surg. 169: 551, 1969. Burch, G. E., and Giles, T. D.: Angle of Traction of the Papillary Muscle in Normal and Dilated Hearts: A Theoretic Analysis of its Importance in Mitral Valve Dynamics, Am. Heart J. 84: 141, 1972. Clark, R. E., and Butterworth, G. A. M.: Characterization of the Mechanics of Human Aortic and Mitral Valve Leaflets, Surg. Forum 22: 134, 1971. Wieting, D. W., Hwang, N. H. C., Kennedy, J. H., and Ruark, B. S.: Engineering Evaluation of Chordae Tendineae and Mitral Valve Function, New York, 1971, American Society of Mechanical Engineers, Publication 71-WA/BHF-4.
o
x (s)
= [Sin {~
(S - s) } d,
Volume 65
Design of leaflet valvular prostheses
Number 6
895
June, 1973
"Normal" valve solution
.5
-1.0
Fig. 8. Coordinates a" and b, as functions of T"/P for the aortic leaflet of the mitral valve for S" = 2.0 em.
1.5 °m,bm 1.0
"Normal" valve solution
0.5 I
as
I
1.0
2,0
3.0
Tm/p in em.
-0.5
-1.0
Fig. 9. Coordinates am and b., as functions of T m/P for the mural leaflet of the mitral valve for Sm 1.0 em.
=
y(s) = [cos
j~
(S - s)} d.
(12)
With the upper limits set equal to S, these integrals gave a and b, the coordinates of the coaptation point (Fig. 3):
a=~{I-cOS~S} b
T
P
P
T
= - sin .- S.
Solutions. I. Existing anatomic datal, 2 provided ranges for the lengths of S" (2.0 em.) and Sm (1.0 cm.) (leaflet width at the midplane) and also for the sum of a" + am (2.0 em.) (the minor diameter of the leaflet during systole). The distances b, and b m (from the plane of the annulus to the coaptation point) were estimated. The geometric conb m was imposed. A leaflet thickstraint that b. ness of 0.5 mm. was specified. 2. Computer solutions used an approximate, trial-and-error method. With the leaflet width S. and Sm specified, the lengths an, am, b., and b m were reduced to functions of the two ratios T./P and Tm/P given by Eq (13) and (14). The four unknowns were tabulated with the TIP ratios as arguments. Finally, a set of lengths was selected
=
(13)
A completely similar set of equations was applied to the second leaflet. The two sets of parameters (T, S, a, b, and eo) were distinguished by a subscript a for the aortic leaflet and m for the mural leaflet.
The Journol of
896
Clark and Sutera
Thoracic and Cardiovascular
Surgery
Table II Coaptation point from annular plane (mm.) Leaflet length projection on annular plane (em.) TIP ratio (em.) Lineal tensions (dynes/em.) Initial slope (degrees) Tensile S (dynes/sq. em.) Chordal tensions (dynes)
from the data plots compatible with the two geometric constraints: b. + b m and a. + am specified number (Figs. 8 and 9). Once obtained, T. and T m were calculated by division of the peak transvalvular gradient (assumed to be 130 mm. Hg).
=
Aortic leaflet
Mural leaflet
2.4 1.37 0.72 1.23 x 10"
2.4 0.72 0.40 0.68 x 10" - 53 13.6 x 10" 2.0 x \0"
- 69
24.6 3.7
x X
10" 10 5
The leaflet profile was then obtained from Eq (9) and the initial slopes 8 and 8 from Eq (8). 3. These computations gave the data shown in Table II. Similar data were obtained for the two abnormal cases. 0 "
0 m
Wth
the gradual but steady improvement in biomaterials, attention has once again turned to exploring the possibilities of a leaflet type of prosthetic valve. The desirability of a central-flow device which avoids the turbulence caused by central occluding mechanisms has long been understood. However, the major concern with leaflet prostheses has been the durability of such devices. Previous laboratory and clinical trials used leaflet prosthetic valves designed on an emperic basis. Long durability of a flexing device can only be achieved by knowledge of the stresses within the normal valve leaflet. No such analysis has appeared to date although experimental stress-strain data for the valves of both man and dog have been published. One method currently under investigation is the use of close-range stereophotogrammetry to obtain the surface character in global coordinates, from which the stresses may be obtained by complex computer processing. Although extremely accurate, the method is time consuming and expensive. As an alternative, classic matheFrom the Division of Cardiothoracic Surgery, Department of Surgery, and the Department of Mechanical and Aerospace Engineering, Washington University, St. Louis, Mo. 63110. Supported in part by U. S. Public Health Service Grants HL 13803-02 and HE 12839-03. Received for publication March 12, 1973.
890
matic techniques were applied to the leaflets of the mitral valve and matched to existing experimental data. This is a report of the analysis. Methods Assumptions. A wide spectrum exists with respect to size and shape of the normal human mitral valve. Therefore, a general analysis was used to accommodate this spectrum based on plausible assumptions. The calculated stresses within any leaflet will depend on the morphologic details and actual dimensions. The first assumption was (I) that the shape of the annulus during peak systole was that of an indented ellipse similar to that of the kidney and (2) that the primary curvature of the leaflets was in a direction across the valve (Fig, I), i.e., transverse to the line of leaflet coaptation. This assumption required that the curvature of the plane of the long axis of the annulus be negligible. This simplication could be justified only if the junctional zones of tissue at the commissures were ignored and the analysis was confined to an idealized infinitesimal strip in the mid-zone of the valve. Unless the commissural zones have significantly greater curvature, the three-dimensional character should not account for an increase of an
Volume 65
Design of leaflet valvular prostheses
Num ber 6
89 1
June. 1973
Left Atrium
/
Annulus
Aortic
I
Mural
CHORDAE TENDINEAE FORCES
=====:;:t dl Leaflet Leaflet
t
Ventricle Fig. 2. Idealized cross-sectional view of the human mitral valve at strip dl. The components of the tension exerted by the chordae tendineae are assumed to be normal to the plane of the annulus.
•
a
----....,.~I
1 I
I I
I I I
I
Fig. 1. View of a normal human mitral valve from the atrial side. The analysis was confined to an idealized infinitesimal strip (dt) in the mid-zone of the valve.
order of magnitude in stresses in those zones. The second assumption concerned the state of stress in the leaflets. If the leaflets are very thin and compliant, and if the minimum radius of curvature under load is much greater than the thickness of the leaflets, than shear stresses and bending moments in the leaflets can be neglected. Correspondingly, the bending moments in the leaflets were assumed to be negligible. These are the classical assumptions in the analysis of stress in very thin shells or membranes. As a consequence , the state of internal stress is one of pure tension which is uniform across the thickness of the membrane. The third assumption was (1) that the forces exerted by the chordae tendineae can be represented as a distributed tension along the free edges of the leaflets and (2) that the component of this tension on the crosssectional plane (Fig. 2) was approximately normal to the plane of the annulus. The annular attachment of the leaflets was visualized as a mechanically ideal hinge, i.e., a
Pressure Loading p. Py - Po
CadI y Fig. 3. Free -body diagram of strip dl. P. , Ventricular pressure . P., Atrial pressure. For other abbreviations, see Appendix.
hinge which offers no resistance to rotation. Finally, the instantaneous force balance at the peak of systole was treated as quasistatic, and inertial forces were neglected . This was justified on the grounds of small leaflet mass and /or negligible acceleration at the instant of peak ventricular systole. Stress analysis. The mathematics of the stress analysis may be found in the Appendix. A free-body diagram (Fig. 3) of a cross section from an infinitesimally thin strip of each leaflet was constructed. Equal lengths of leaflet coaptation were assumed . Assuming a static equilibrium on an arbitrary differential element of the membrane (Fig. 4) at given left ventricular and left
The Journal of
892
Clark and Sutera
Thoracic and Cardiovascular Surgery
dll ..
00
= 1.42 em. - - - - - I
So = 2.0em
P = 130 mm Hg
y Fig. 4. Representation of static equilibrium on an arbitrary differential element of the membrane.
1 - - - - a., 1 . 3 7 e m . -
.
am' 0.72 em.
Fig. 6. The geometry of the human mitral valve in acute left ventricular dilatation.
aortic
Sm· I . O em
So = 2.0 em P
= 130 mm Hg
p=
130 mm Hg
Fig. 5. The geometry of the idealized normal human mitral valve.
Fig. 7. The geometry of a ballooning mitral valve in the floppy valve syndrome.
atrial pressures, it was possible to calculate the internal forces (force per unit length) by developing a series of ordinary differential equations. These first led to the conclusion that the magnitude of the tension within the leaflet was constant along the entire strip taken. It is important to note that the leaflets were assumed to have a curved shape with the valve fully closed. Existing anatomic data for leaflet lengths and coaptation lengths were used. The multiple equations were solved by trial-and-error analysis with the use of a computer. Two additional cases were then considered by means of the same techniques. These were (I) acute left ventricular dilatation and (2) the "floppy" or ballooning mitral valve. In the first case, the primary change was assumed to be in
the long axis of the left ventricle; hence, the leaflets were pulled down into the ventricle, significantly altering the shape of the mitral valve at closure. In the second condition, the floppy valve, only changes in the lengths of the transverse leaflet and of the chordae tendineae were altered. A leaflet thickness of 0.5 mm. was assumed in all cases. These differences are compared in Figs. 5 to 7. The results of the analysis are summarized in Table I. Since the tension was found to be equal throughout the strip analyzed, the total chordal force was calculated by multiplying by an assumed leaflet length. The papillary muscle forces were calculated by adding one half of each total leaflet force. These computations allowed comparison to published experimental data."
Volume 65
Design of leaflet valvular prostheses
Number 6
893
June, 1973
Table I. Stresses in human mitral valves
Valve condition
Normal Acute left ventricular dilatation Floppy valve syndrome
Aortic leaflet Mural leaflet (in 10 5 dynes/ (in 10 5 dynes/ sq. cm.) sq. em.)
24.6 33.2 21.0
Discussion Continued efforts toward the development of a central-flow device for the mitral position seem justified because of functional hydrodynamic deficiencies of prostheses." To this end, appropriate guidelines must be developed for both the valvular design and the choice of materials for a central-flow device. Therefore, an analysis of the lineal tensions, leaflet stresses, and total forces encountered by the chordae tendineae is required for the normal human mitral valve. The data derived from the present analysis were compatible with previous stress-strain studies," which indicated that the valve operates in a range of anisotropic and then isotropic behavior. Furthermore, the data of Wieting and colleagues, G derived from an experimental study of normal human valves, showed that the summed forces for the papillary muscles in normal valves were 7.60 x 10" dynes." The value derived from our studies is 5.7 x 10" dynes for the normal component of total chordal tension, calculated from assumed values of valvular dimensions from the literature. Small adjustments of the assumed leaflet edge lengths and/or a traction-angle factor could result in exact matching of the experimental data. The two additional cases considered are interesting because of the clinical analogies and because they provide a frame of reference for the normal case. Further, the marked difference in stress found for the condition of acute ventricular dilatation demonstrates that the geometry of the valve leaflet is highly important with respect to the stresses sustained.
Normal component of chordal tension, each leaflet (10 5 dynes)
13.6 36.0 9.6
Aortic
I
Mural
Normal component of total chordal tension (in 10 5 dynes)
3.7
2.0
5.7
5.0
5.4 1.4
10.4
3.2
4.6
In left ventricular dilatation, the summed forces carried by the chordae tendineae were nearly twice those of the normal valve (93 per cent increase). It is unlikely that the chordae tendineae would rupture, even with marked distortion of the mitral valvular apparatus in this condition. However, in the presence of previous disease (subacute or acute bacterial endocarditis) and/or concurrent disease of the collagen matrix of the chordae, a significant increase in tensile stress could conceivably cause rupture. Also, the possibility of a simultaneous increase in the traction angle, as proposed by Burch and Giles, I would aggravate the resultant stress levels in the individual chordae. The magnification factor is a cosine function and results in increased values for chordal loads compared to the load calculated in this report.
Summary Well-known mathematic techniques, comprising membrane theory and computer analysis, have been used to calculate the internal stresses at peak ventricular systole in the aortic and mural leaflets of the normal mitral valve and in two cases of altered geometry-acute left ventricular dilatation and the floppy mitral valve. The results demonstrate that, for the assumptions made and the dimensions used, the calculated peak stresses correspond closely to those found experimentally for the normal valve. Further, calculations for simulated pathologic clinical conditions showed that the stresses in the leaflets are nearly doubled in the case of acute left ven-
894
The Journal of Thoracic and Cardiovascular Surgery
Clark and Sutera
tricular dilatation but are reduced in the case of the floppy mitral valve. This study has aided the design and selection of suitable materials for use in prosthetic leaflet heart valves.
and dT d8 -tanfl+T--p ds ds
2 3
4
5
6
dT (cot fl + tan 8) = O. ds This can only be satisfied for any 8 if dT
-
C:.
it.
ments and at the free edge be and These vectors are directed along the local tangents to the membrane curve. 2. The profile of the membrane curve will be described in terms as a function of 8(s) where s is the curvilinear coordinate (Fig. 3). 3. Taking an element length (Fig. 4) ds, located at point Q, forces T(s) and T(s + ds) act on the element and represent the internal tensions exerted by neighboring elements. 4. Static equilibrium of the element requires the sum of the instantaneous forces to be zero. Therefore ~
F,
=
0,
~
r,
=
= O.
(4)
constant.
(5)
ds
Therefore
T Therefore d8
P
ds
T
(6)
Integration of Eq (6) yields
e
(s)
Ps
:-= -
T
+ 8
0
)
7r
fl = - at s 2
S.
Thus 8
0
7r
P
2
T
=- -
S.
P(S - s).
2
dT cot ds
e -
d8 T - + P ds
o
(2)
(9)
T
Since dy ds
dx = Sin 8, - = Cos fl ds
(10)
Integration yields the x and y coordinates x (s) =JCos 8 ds, y (s) =JSin 8 ds. (11)
Substitution of Eq (9) into (11) yields
and
(8)
Eq (7) now becomes
(1)
0
(7)
0 •
This angle (8 was eliminated in favor of the total strip length S, measured along the strip from the annulus to the coaptation point by inserting the condition
Appendix Formulation. I. Let the lineal tensions at the annular attach-
(3)
adding Eq (2) and (3)
REFERENCES Rusted, I. E., Scherfley, C. H., and Edwards, J. E.: Studies of the Mitral Valve. I. Anatomic Features of the Normal Mitral Valve and Associated Structures, Circulation 6: 825, 1952. Chiechi, M. A., Lees, W. M., and Thompson, R.: Functional Anatomy of the Normal Mitral Valve, 1. THoRAe. SURG. 32: 378, 1956. Nolan, S. P., Stewart, S., Fogarty, T. J., Dixon, S. H., Jr., and Morrow, A. G.: In Vivo Studies of Instantaneous Blood Flow Across Mitral Ball-Valve Prostheses: Effects of Cardiac Output and Heart Rate on Transvalvular Energy Loss, Ann. Surg. 169: 551, 1969. Burch, G. E., and Giles, T. D.: Angle of Traction of the Papillary Muscle in Normal and Dilated Hearts: A Theoretic Analysis of its Importance in Mitral Valve Dynamics, Am. Heart J. 84: 141, 1972. Clark, R. E., and Butterworth, G. A. M.: Characterization of the Mechanics of Human Aortic and Mitral Valve Leaflets, Surg. Forum 22: 134, 1971. Wieting, D. W., Hwang, N. H. C., Kennedy, J. H., and Ruark, B. S.: Engineering Evaluation of Chordae Tendineae and Mitral Valve Function, New York, 1971, American Society of Mechanical Engineers, Publication 71-WA/BHF-4.
o
x (s)
= [Sin {~
(S - s) } d,
Volume 65
Design of leaflet valvular prostheses
Number 6
895
June, 1973
"Normal" valve solution
.5
-1.0
Fig. 8. Coordinates a" and b, as functions of T"/P for the aortic leaflet of the mitral valve for S" = 2.0 em.
1.5 °m,bm 1.0
"Normal" valve solution
0.5 I
as
I
1.0
2,0
3.0
Tm/p in em.
-0.5
-1.0
Fig. 9. Coordinates am and b., as functions of T m/P for the mural leaflet of the mitral valve for Sm 1.0 em.
=
y(s) = [cos
j~
(S - s)} d.
(12)
With the upper limits set equal to S, these integrals gave a and b, the coordinates of the coaptation point (Fig. 3):
a=~{I-cOS~S} b
T
P
P
T
= - sin .- S.
Solutions. I. Existing anatomic datal, 2 provided ranges for the lengths of S" (2.0 em.) and Sm (1.0 cm.) (leaflet width at the midplane) and also for the sum of a" + am (2.0 em.) (the minor diameter of the leaflet during systole). The distances b, and b m (from the plane of the annulus to the coaptation point) were estimated. The geometric conb m was imposed. A leaflet thickstraint that b. ness of 0.5 mm. was specified. 2. Computer solutions used an approximate, trial-and-error method. With the leaflet width S. and Sm specified, the lengths an, am, b., and b m were reduced to functions of the two ratios T./P and Tm/P given by Eq (13) and (14). The four unknowns were tabulated with the TIP ratios as arguments. Finally, a set of lengths was selected
=
(13)
A completely similar set of equations was applied to the second leaflet. The two sets of parameters (T, S, a, b, and eo) were distinguished by a subscript a for the aortic leaflet and m for the mural leaflet.
The Journol of
896
Clark and Sutera
Thoracic and Cardiovascular
Surgery
Table II Coaptation point from annular plane (mm.) Leaflet length projection on annular plane (em.) TIP ratio (em.) Lineal tensions (dynes/em.) Initial slope (degrees) Tensile S (dynes/sq. em.) Chordal tensions (dynes)
from the data plots compatible with the two geometric constraints: b. + b m and a. + am specified number (Figs. 8 and 9). Once obtained, T. and T m were calculated by division of the peak transvalvular gradient (assumed to be 130 mm. Hg).
=
Aortic leaflet
Mural leaflet
2.4 1.37 0.72 1.23 x 10"
2.4 0.72 0.40 0.68 x 10" - 53 13.6 x 10" 2.0 x \0"
- 69
24.6 3.7
x X
10" 10 5
The leaflet profile was then obtained from Eq (9) and the initial slopes 8 and 8 from Eq (8). 3. These computations gave the data shown in Table II. Similar data were obtained for the two abnormal cases. 0 "
0 m