Dimensions and Geometric Relationships of the Human Aortic Value as a Function of Pressure

Transcription

1 Dimensions and Geometric Relationships of the Human Aortic Value as a Function of Pressure By W. Milton Swanson and Richard E. Clark ABSTRACT In a continuing effort to develop improved prosthetic heart valves, a redefinition of the anatomy of the human aortic valve as a function of stress was undertaken. Dimensions and geometric relationships of the human aortic valve as a function of intraaortic pressure between and 1 mm Hg were obtained from a series of silicone rubber valve casts. The axial length of the valve region was found to vary negligibly with pressure, but significant variations in geometry and angular dimensions were seen. The leaflet attachment annulus forms an ellipse at the plane of intersection with the cylindrical surface passing from the left ventricular tract through the aorta. Deductions from stress considerations for the measured geometry indicate that the loaded leaflet is a section of a cylindrical surface. The equation for this developed surface was obtained, and a prosthetic design was determined using average values at 1 mm Hg. The leaflet is developable onto a plane with a cut required along part of the junction line between the initially cylindrical part and the plane coapting surfaces. Optimum valve shape mandates a base angle between the cylindrical leaflet and the center axis of 7 (a =-22, where a is the leaflet angle). KEY WORDS aortic valve structure leaflet shape and dimensions prosthetic valve design aortic modulus stresses in valve leaflets Downloaded from by on January 2, 19 An accurate definition of the geometry of the aortic valve is necessary prior to development and fabrication of a prosthetic valve. As part of a program to determine the geometry and structure of the human aortic valve, silicone rubber molds were cast under pressure. Measurements considered to be important were made and analyzed. Preliminary studies in our laboratory have demonstrated the sensitivity of in-plane stresses to the geometry of this structure during diastole and systole (1). Previous investigations by Wood et al. (2) and Sauvage et al. (3) utilized pig hearts and a freezing technique under pressure. Recently, Mercer et al. (4) have investigated the geometry of the human aortic leaflet via a molding technique at 1 mm Hg of pressure. The present paper is a report on our 2-year investigation of the geometry and proportionalities of the human aortic valve from which important design conclusions can be drawn. Accurate knowl- From the Departments of Mechanical Engineering and Cardiothoracic Surgery, Washington University, St. Louis, Missouri This work was supported in part by U- S. Public Health Service Grant HL-1383 from the National Heart and Lung Institute. Received January 31, Accepted for publication August 8, Circulation Research, VoL 35, December 197k edge of valve and sinus region geometry is required for flow calculations yielding information on leaflet motion during opening and closing (5). Methods Fresh human hearts were obtained at autopsy, stored at 4 C, and used within 1-3 days after death. The specimens consisted of two to three diameters of aorta beyond the sinuses of Valsalva and one-half to one diameter of tissue on the left ventricle side. The aorta was held with three hemostats hung on ring-stand hook arms. Then, 4-5 ml of low-viscosity room temperature-vulcanizing silicone rubber (RTV GE-11) was prepared. The coronary arteries were at first tied off, but it was later found that coronary leakage could best be eliminated by plugging them with silicone rubber beads, 4-5 mm in diameter. Part of the silicone rubber was injected into the sinus pockets with a -ml syringe and a 6-cm tube extension to allow filling from the bottom up to eliminate air pockets. When the preparation was nearly full, a grooved stopper with a 5-cm length of glass tube in it was slowly pushed into the aorta, filling the tube. The aorta was secured to the stopper with umbilical tape around the groove. The remaining silicone rubber was poured into a large reservoir syringe connected to the stopper tube with a short piece of flexible tubing, and the reservoir syringe was then suspended on a ring stand. A tube through a stopper in the top of the reservoir syringe was connected through a T-tube to a 871

2 872 SWANSON, CLARK Downloaded from by on January 2, 19 standard sphygmomanometer. The pressure was gradually increased to the desired casting pressure and maintained during the cure. A jar of saline was placed around the aorta to maintain a moist condition and a temperature of 37 C. This slightly warm temperature also accelerated the cure. A period of about 2 hours was required for a minimum stable-dimension cure. The cast was removed, and the procedure was repeated at the next pressure. With this technique, five or six casts could be made in 1 day. The deteriorating effect of the casting procedure on the elastic properties of the aorta was determined by repeating the preparation of a cast at mm Hg. This check cast was made on three series (series 5 and 6 of Table 2 and one other) after the last cast at maximum pressure had been made. No significant dimensional variations were found. One series was also repeated at 5 mm Hg, and no significant variations were noted. The series 8 casts are shown in Figure 1. Data on subject aortas are presented in Table 1. The significant dimensions recorded in Tables 2-4 were measured with vernier calipers to the nearest.1 mm. In some critical cases, several sets of readings were taken for one dimension and averaged to determine repeatability. The variations obtained were usually within.2 mm or about 1%. Dimensions involving the three separate sinuses and leaflets were average for the three. The noncoronary sinus was usually, but not always, the smallest. A profile tracing of the sinus region in a plane perpendicular to the center axis was made and planimetered to obtain the maximum sinus area from _yvhich an equivalent area-averaged diameter, d,, was determined. Each cast tracing was planimetered ten times to get an accurate meas- TABLE 1 Valve Origin Valve series Age (years) Sex T» M F M F F M 4f urement. The ten measurements usually did not vary by more than 1%. When the variation was larger, more readings were taken. The circumscribing sinus diameter, dsm, was also recorded. Nomenclature c d E Ed f h I p x, y a <f> a = C o a p t a t i o n (Fig. 2). = D i a m e t e r (mm) (Fig. 2). = E l a s t i c m o d u l u s (dynes/cm*). = E l a s t i c m o d u l u s based on a o r t i c d i a m e t r a l s t r a i n : E,, = Ap/Ad a /d a ) ( d y n e s / cm 2 ). = F r e e e d g e (cm) (Fig. 2). = Height from ventricular tract base plane to top of annulus fibrosis (Fig. 2). = Length (cm) (Fig. 2). = Pressure (dynes/cmj). = Coordinates. = Leaflet angle (Figs. 2, 9). = Angles (Fig. 9). = Free edge angle (Fig. 2). = Stress (dynes/cm2). FMURE 1 a: Series 8 valve molds, to 1 mm Hg. b: 8-mm Hg mold mated with left ventricle mold. Circulation Research, VoL 35, December 1971,

3 AORTIC VALVE DIMENSIONS AND GEOMETRY 873 SUBSCRIPTS a = c m = s = V = Aorta. Center. Maximum. Sinus of Valsalva. Ventricular tract. SUPERSCRIPTS * = Dimensionless quantity referred to d v. # = Dimensionless quantity referred to that quantity at zero pressure. Results The valves listed in Tables 1-4 were chosen for evaluation. Previous series were developmental and had too few casts to yield significant data. The dimensions obtained from cast measurements are listed in Tables 2-4. Figure 2 illustrates the valve region and shows representative dimensions. Graphic illustrations of dimensionless variations with pressure are presented in Figures 3-5. Dimensions were reduced to dimensionless variables (denoted by asterisks) with respect to the diameter, d v, of the left ventricle immediately below the aortic valve for two reasons: d v is the flow inlet diameter, and it varies very little with pressure. The average variation of d v with pressure from mm Hg to 1 mm Hg was approximately 1% (Fig. 3). For calculating the flow conditions through the valve region (5), the sinus dimensions d^*, d sm *, and ^s* are significant along with the leaflet length c * and the center and maximum coaptation dimensions Q* and Cm*. d v was measured as the diameter of an indentation ring formed at the annular attachment (Fig. 2). The inlet flow diameter is slightly smaller than d v by the thickness of Dimensional Quantities TABLE 2 Series P (mm Hg) (1) d v (mm) (2) d. (mm) (3) h (mm) (4) <t> (5) a (6) E,, x 1-3 (dynes/cm 1 ) Downloaded from by on January 2, Average Average at 1 mm Hg See text for abbreviations. Circulation Research, Vol. 35, December I97i

4 874 SWANSON, CLARK Dimensionless Quantities Relative to Inlet Diameter TABLE! Series P (mm Hg) (1) d,* (2) h* (3) d,* (4) d*=* (5) (6) t>* (7) c,* (8) c * (9) f* Downloaded from by on January 2, Average Average at 1 mm Hg See text for abbreviations the leaflet at its base and by the restriction of the valve at its maximum opening (3) (broken leaflet line in Fig. 2). The dimensions shown in Figure 2 are for a representative valve; they were determined from averages of mold measurements of the illustrated quantities. ^c. is the length measured along the leaflet at the center of the sinus region and is the center coaptation plus the bottom face length STRUCTURAL OBSERVATIONS Dimensions of significance with respect to valve structure are shown in Figure 2. The height, h, from the bottom of the attachment to the top of the commissure varied only slightly with pressure (Fig. 5). The largest dimensional variation with pressure was for the sinus diameters dj and d m. Column 6 of Table 2 gives the elastic modulus based on the aortic root diameter (d a ) variation (re- FIQUHE 2 Geovietry and relative dimensions of aortic valve region. See text for abbreviations. Circulation Research, VoL 35, December I97h

5 AORTIC VALVE DIMENSIONS AND GEOMETRY 875 TABLE 4 Dimensionless Quantities Relative to Zero Pressure Value Downloaded from by on January 2, 19 Series P (mm Hg) (1) d v * See text for abbreviations. (2) d.' ferred to as pressure modulus in ref. 6: E d = Ap/(d a /d a ). The range 1.6 x 1 5 < E rt < 5.3 x 1 5 dynes/cm 1 in Table 2 is in the range of published data for pig aortas (E d = 2 x 1 s dynes/cm 2 [6]) and for the femoral artery (2 x 1 5 < i^ < 6 x 1 6 dynes/cm 2 [3, 7]). This relatively large range of values for a physiological property is not unusual. The modulus determined in this manner might include inaccuracies because of the method of determining the strain Ad a /d a. The uncertainty for E d varied from about 1% at mm Hg to about 1% at 8 mm Hg. The maximum local data variation for E d values calculated from a smoothed curve of Ap vs. (d a * - 1) was 15%. More accurate means determined using special strain testing apparatus (8) give true moduli and yield values of the circumferential modulus E for the aorta of about 4 x 1 6 dynes/cm 2 at 1% strain. Values of E d were converted to E by multiplying by d/2t, which was about 1 (where t is the wall thickness). The average resulting E of approximately 5 x 1 s dynes/cm 2 was close to published data Circulation Research, VoL 35, December 1S7J, (3) h* (4) d.' (5) d. n ' (6) fr* SYMBOL o D A (7) e, * SERIES m Y 2L m VTTT PRESSURE-p (mm Hg) FIGURES Relative dimensional variation of inlet diameter, djwith pressure, p.

6 876 SWANSON, CLARK Downloaded from by on January 2, LLJ I- I Zx LU H- uo s"2 LU.8 h PRESSURE-p (mm Hg) FIGURE 4 Dimensionless leaflet center length, t r *, as a function of pressure, p. (8) for this type of comparison. This modulus is not constant but increases with increasing load. The marked nonlinear behavior of aortic leaflets gives values of E from 2 x 1 5 to 6 x 1 7 dynes/cm 2 (9, 1). The width of the coaptation at the center, c,., decreased with increasing pressure as a consequence of a greater increase in diameter than in length. The coaptation surface then "peeled back" as the diameter of the leaflet supporting structure increased with increasing pressure. Also, because of the small rate of increase of d v with pressure, the sinuses bulged out over this base diameter producing an increase in the angle of the lower leaflet surface, a, and a decrease in the angle of the free edge, <f>, as indicated in Table 2 (columns 5 and 4, respectively) and in Figures 6 and 7. The free edge length, f, increased only slightly at the expense of a decrease in coaptation width (c,. and, and it increased as a increased and decreased (i.e., d a increased more than d v ). This behavior was also evident from sections through one of the sinuses at three different pressures (Fig. 7a). Sauvage et al. (3) also indicated a slight increase in f with pressure; their dimensionless results (3, Figs. 1-14) are quite similar to ours and give f* =.58 at 1 mm Hg for pig hearts compared with f* =.62 for human hearts. They indicated that < decreases from 34 to 24 when pressures are increased from 8 mm Hg to 1 mm Hg with <f> = 28 at 1 mm Hg compared with our value of =32 at 1 mm Hg. These results are as close as can be expected considering the difference of species. The most obvious significant structural or geometric difference is in the configuration of the sinuses of Valsalva (6, Figs. 1-6). In addition to the difference in species, freezing also produces a variation effect on tissue properties (9). Uncertainties based on errors from repeated measurements of basic quantities were 5% for f*, <j>, and c*, 4% for a, 3% for <? c *, and 2% for other dimensionless quantities. These values are close to the maximum relative variations of data points from smoothed curves. Since the leaflets meet at an angle at the noduli Aranti and since the pressure loading is balanced across the coapting surfaces, there can be no stress along the free edge or in the coapting surfaces in the central region except for the compressive stress equal to the pressure. LEAFLET STRUCTURE Striations on the surfaces of the casts adjacent to the leaflets indicated a fibrous structure across the leaflet on the aortic ' PRESSURE -p (mmhg) FIGURES Dimensionless overall vertical height, h*, as a function of pressure, p. Circulation Research, Vol. 35, December 197U

7 AORTIC VALVE DIMENSIONS AND GEOMETRY 877 o z 6 u 4 (b) Downloaded from by on January 2, PRESSURE-p (mm Hg) FIGURE a Bottom leaflet surface angle, a, variation with pressure, p. surface (Fig. 8). These striations were in a plane perpendicular to the axis of the cylindrical leaflet surface and extended from one attachment to the other. Mating casts in the left ventricle side had a smooth surface adjacent to the leaflets. The leaflets are essentially thin flexible membranes, and they tend to form a cylindrical surface between their points (or lines) of main support. Sections through the leaflet profiles along the striation lines are shown in Figure 7b. Since the leaflets end at a free edge in a section through the coaptation zone, there can be no radial stress component in them. This conclusion is also corroborated by the fact that the radial profiles were essentially straight (no significant definable curvature in the radial direction [Figs. 8 and 9]). The only load stress component is then the circumferential stress carried by the circumferential collagen fiber structure. LEAFLET THICKNESS Leaflet molds were made on the series 8 aortic molds. The fibrous structure in the cylindrical portions and in the coapting surfaces closely resembled that indicated in Figures 8 and 1. As the molding pressure was increased, the coapting leaflet thickness decreased. Measurements made on series 8 casts gave a 3% decrease in average thick- Circulation Research, VoL 35, December 1971, FMURE7 a: Sections through a sinus vertical center plane at SO, 6, and 8 mm Hg. b: Cylindrical sections at SO, 5, and 8 mm Hg. Tick marks indicate leaflet attachment points. ness measured at the midpoint of the coapting surfaces from.48 mm to.32 mm (Fig. 11). Variations were large from one leaflet to another on the same valve at a given pressure. At 1 mm Hg, thickness varied from.22 mm to.4 mm. OVERALL STRUCTURE The valve structure consists of thin flexible sheets (the leaflets) freely suspended between the attachments, forming interleaflet seals along the coaptation zone. Details of an idealized valve structure are shown in Figure 9. Figure 9a is a view looking from the left ventricle side. The loadcarrying collagen fibers appear in the angled view of the bottom side of the leaflet as ellipses. The attachment annulus line which forms the three-way intersection of the leaflets with the sinus and ventricular tract walls projects into a circle (the left ventricular outlet tract diameter) in this view. A section in the plane of the circular arc through the leaflet is shown in Figure 9c. The leaflet contour b in Figure 9c is onethird of a circle. The adjoining sinus contour is also nearly circular. The leaflet and sinus curvatures are parallel at their line of attachment intersection with the left ventricular outflow tract wall yielding a load-stress balance, as indicated by the arrows in Figure

8 878 SWANSON, CLARK B ;.. 1 AORTA Endothelium Coorse Collage nous Circumferential Bundles - Radial Collagen and Elastic Fibres Epithelium Circumferential & Radial Fine Fibres VENTRICLE Downloaded from by on January 2, 19 FIGURE 8 A: Sketch showing circular collagen fiber structure of leaflets. B: Photograph of mold surface.c: Sketch of cross section of leaflet. 9c. During diastole, there is no pressure loading on the left ventricular outflow tract wall; consequently, it can carry no significant stress component to balance a directional component of the leaflet and sinus wall stresses. In the projection of Figure 9b, the attachment line is straight. The intersection of this plane surface of the attachment intersection with the cylindrical contours of the leaflet then necessarily indicates that the attachment line is elliptical. This same conclusion is obtained considering that the attachment line is also the intersection of its plane with the cylindrical inflow section (dv). The leaflet is then a one-third section of a cylinder. Some authors have represented this contour as being parabolic (4). From observations that (1) the attachment line forms a plane and intersects with the nearly cylindrical surface formed by the left ventricular outflow tract and the aorta and (2) the leaflet contour is cylindrical, it is concluded that the angles /3 and y in Figure 9b must be equal and furthermore, that the leaflet cylinder diameter must equal dv (verified by direct measurement). From Table 2, the average value of the angle a (at 1 mm Hg) is 22 ; therefore, /3 =y =(9 -a)l2 = 34. Starting with the profile of Figure 9b as shown in Figure 1a and the one-third circle projection (Fig. 1b corresponding to Fig. 9a), the one-third circle leaflet contours of Figure 1c (corresponding to Fig. 9c) result if y =/3. Figure lod shows the elliptical projection of the attachment line profile. Since the basic geometry originates from a one-third circle section, the elliptical contour of Figure lod is not a complete ellipse: its ends are not parallel and the contour could easily be mistaken for a parabola. Since the elliptical contour of Figure lod is formed with the intersection of only one-third of a circle, its length relative to the major axis of the full elliptical contour is 1/2, and the width relative to the minor axis is V3/2. The major axis-minor axis ratio is cosecant y = The width of the elliptical Circulation Research, Vol. 35, December 19?i

9 879 AORTIC VALVE DIMENSIONS AND GEOMETRY tany FIGURES Downloaded from by on January 2, 19 Views of aortic valve, a: View from left ventricle, b: Side view iv plane of attachment line, c: Section through contour b of Figure Ha. attachment line section is just the one-third chord of the dv circle or.866 relative to dv. The relative length of the elliptical contour is 1/4 of the major axis length or.45. The contours of the attachment annulus come together at the top in a vertical short commissure section as indicated in Figure 1a, c, and d. The center coaptation also turns up to the vertical (Fig. 1a). The platform projection of the leaflet in Figure loe is obtained from laying out chord lengths on the projections from 1a as calculated from arc lengths from 1c. The leaflet contour chord length is obtained from the cylindrical leaflet surface contour in Figure 1c and is just x=dv/2 or x*=6/2, where is the angle out from the center along the leaflet arc whose radius is dv/2. The longitudinal coordinate obtained from projection onto the plane perpendicular to the axis of the leaflet is V _ dr(l - cosfl) 2tany Substituting 6 = 2x* into this expi-ession for y gives Circulation Research, Vol. J.5. December 1971, for the equation of the leaflet contour. Figure lof is a layout of the vertical coapting plane surfaces as the projection from Figure 1a and b. The bottom curves are parts of ellipses formed by the intersection of the coaptation planes with the leaflet cylinder section. The two coapting planes from Figure lof are reconstructed on Figure loe to be coincident at the commissure attachment point and tangent to the top curve of the cylindrical leaflet section there. The light internal lines of Figure 1 are representative of the collagen load-carrying fiber bundles. The load in the top point of the cylindrical section (Fig. loe) is carried by the fibers running down through the coaptation zones (two fiber lines are illustrated). The parts of the coaptation surfaces above the lines to the point are unstressed (except for the compressive pressure loading). The entire geometry is essentially determined by the angle a (since (3 = y). The commissure height (.37) and the center coaptation height (.17) do not affect the final shape of the cylindrical part of the leaflet or its load-carrying ability and stress. The size of the split necessary to allow a development of the coapting surfaces with the cylindrical surface with contiguity at the attachment lines shown in Figure loe is also determined by a. The value < a < 25 minimizes this separation and the size of the slit window where the fold line diverges. If the leaflet thickness were uniform, the stress would be cr = pd/2t, since it is a thin membrane (t < < d) with a uniform radius (cylindrical section). The maximum value of the pressure, p, is about 1 mm Hg at valve closure. The maximum membrane stress is then on the order of 27 x 15 dynes/cm2 for a leaflet thickness of.5 mm. Discussion The design geometry derived from valve cast measurements is an averaged representative geometry for the aortic heart valve. The simple cylindrical geometry of the load-carrying part of the leaflet gives a uniform stress resultant (load per unit thickness) equal to pd/2. This derived simplified geometry does not take into account variations from one leaflet

10 88 SWANSON, CLARK Downloaded from by on January 2, 19 FIGURE 1 a: Side -profile through attachment plane, b: View from left ventricular side, c: Section through cylindrical leaflet profile, d: Projected attachment line profile, c: Developed surface of leaflet, f: Planar layout of coapting surfaces. Dimensions are relative to d r * = PRESSURE-mm Hg FIGURE 11 Coapting leaflet thickness as a function of pressure. to another, specifically observed variations in attachment line geometry between coronary and noncoronary leaflets. These variations were not included because of the objective of obtaining a simplified geometry that could be fabricated for clinical installation and because the variations were not large enough to be considered as physiologically significant for prosthetic valve installation. The leaflet and attachment load stresses are maximum at valve closure. Bending stresses during the folding wave motion during valve opening are an order of magnitude smaller than the static load stresses following valve closure (11). Since the free edge is always unloaded or unstressed, its length should not change significantly. As the diameter at the top of the commissure increases with pressure, the free Cirathtion Raearch, VoL 35, December 197i

11 AORTIC VALVE DMENStONS AND GEOMETRY 881 Downloaded from by on January 2, 19 edge angle, tf>, should then decrease significantly if f is to remain about constant. These conclusions are verified with reference to the variations of f and <f> with pressure in Tables 2 and 3. The large apparent redundancy of the coapting surfaces above load-carrying fiber lines through the point of the cylindrical section (Fig. 1) serves several purposes. First, this zone actually decreases in area as the pressure is increased as described earlier in this paper. Second, as the leaflets fold back during systole, their length is such that» h so that the free edge of the open leaflets forms a circle providing a flow separation surface of maximum leaflet and flow stability against leaflet vibrations and local flow separations. We also speculate that this geometry in the opening transition and the open state is critical in the development of the circulatory vortex flows in the sinuses. It is to be expected that the circumferential leaflet modulus should be relatively high, since the primary leaflet load-carrying structure is circumferential, coarse, relatively stiff collagen fibers as indicated by the striation patterns on the casts of the aorta (Fig. 8) and by photomicrographs (12). Since there is no significant load in the radial direction, the valve leaflet is less stiff in that direction, allowing the greater flexing motion during opening and closing (5). The casts and photomicrographs also show a smooth, fine fiber structure on the ventricular side. The apparent paradox that the radial modulus is less than the circumferential modulus (9, 1) although the radial length variation with pressure is small is resolved by the condition that the radial stress is negligible. The length along the leaflet, c, measured in a radial plane is then about constant. As the valve diameter increases, the center coaptation, Cc, must decrease to yield a constant c. In fact Ce does decrease, but not enough to yield a constant f c. The f c increases can then be attributed to a very low longitudinal modulus, a consequence of the coarse, stiff, circumferential collagen fibers "rolling" in a less stiff thinner collagen and elastic fibrous matrix material. The leaflet must be quite elastic in the radial direction (9) to maintain the sharp curvature at the coaptation intersection without producing large stresses. The bending stress at the coaptation inter- Cimilation Research, VoL 35, December lbti section line is further decreased by a very thin structure there. The primary leaflet deformation during opening and closing is bending in radial (or meridional) planes (5). The structure just described and the corresponding modulus behavior then minimize these bending deformation stresses. The cylindrical leaflet shape is the simplest constant-stress structure for a uniformly thick, uniformly loaded membrane. If a weakness exists in the membrane, it will bulge to a double curved shape. Since the stress is inversely proportional to curvature, a locally decreased stress results. Observations of a number of the leaflets with significant nonuniformities from one cusp to another also showed significant bulges in the leaflets near their roots at the higher pressure levels of 8-1 mm Hg. The small variations of base diameter, d v, and overall effective valve height with pressure are advantageous with respect to prosthetic valve design and installation. The diameter into which the valve is sutured and the height of the vertical attachment tissue to which the stent may be sutured vary only slightly with pressure. The stresses produced because of the difference in vertical stretch of the sinus-aortic stricture and the more rigid prosthetic valve structure from strains on the order of these variations in d v and h are then small. The large apparent decrease in leaflet thickness with increasing pressure may be a consequence of the highly folded and crimped collagen fiber structure as well as a true compression or water absorption-desorption effect. The grossly nonlinear stressstrain behavior (9) may also be attributed to initial straightening of the crimped and folded collagen fiber structure. The initial low modulus range occurs as the fibers are straightened out; the posttransition modulus exists as the straightened fibers are stressed. This behavior is similar to that obtained with crimped synthetic fibers or with rubber as the coiled molecules are straightened during the pretransition phase and the posttransition phase occurs for deformation of the stretched out molecules. It is also interesting that the projected pressure load produced by 1 mm Hg over the area of d v is about 1 8 dynes (equivalent to a weight load corresponding to about 1

12 882 SWANSON, CLARK kg). This downward force is balanced primarily by the distributed pressure force on the curved wall of the aortic arch. References 1. GOULD PL, CATALOGLU A, DHATT G, CHATTOPA- DHYAY A, CLARK RE: Stress analysis of the human aortic valve. National Symposium on Computerized Structural Analysis and Design, G«orge Washington University, WOOD SJ, ROBEL SB, SAUVAGE LR: Technique for study of heart valves. J Thorac Cardiovasc Surg 46: , SAUVAGE LR, VIGGERS RF, BERGER K, ROBEL SB, SAWYER PN, WOOD SJ: Prosthetic Replacement of the Aortic Valve. Springfield, Illinois, Charles C Thomas, MERCER JL, BENEDECTY M, BAHNSON HT: Geometry and construction of the aortic leaflet. J Thorac Cardiovasc Surg 65: , SWANSON WM, CLARK RE: Aortic valve leaflet motion during systole. Circ Res 32:42-48, MOZERSKY DJ, SUMNER DS: Transcutaneous measurement of the elastic properties of the human femoral artery. Circulation 46: , ARNDTJO, KOBERG: Die Druck-Durchmesser-Beziehung der intakten A. Femoralis des wachen Menschen und ihre Beeinflussung durch Noradrenalin-Infusionen. Pfluegers Arch 318:13-146, MINNS RJ, SODIN PD: Role of the fibrous components and ground substance in the mechanical properties of biological tissues: A preliminary investigation. J Biomechanics 6: , CLARK RE: Stress-strain characteristics of fresh and frozen human aortic and mitral leaflets and chordae tendineae. J Thorac Cardiovasc Surg 66:2-8, CLARK RE, BUTTERWORTH GAM: Characterization of the mechanics of human aortic and mitral valve leaflets. Surg Forum 22: , SWANSON WM, CLARK RE: Motion and stresses in aortic valve leaflet during systole. American Society of Mechanical Engineers Paper 72-WA/BHF-5, CLARK RE, FlNKE EH: Scanning and light microscopy of human aortic leaflets in stressed and relaxed states. Cardiovasc Surg 67:792-84, 1974 Downloaded from by on January 2, 19 Circulation Research, VoL 35, December I97i