RESEARCH NOTE Finite element stress analysis

of a human aortic valve

Min-Tse Chen, M. P. Rossow and P. L. Gould

Department of‘ Civil Engineering, Washington University, St. Louis, Missouri 63130, USA

and R. E. Clark

Department qf’ Surgery and Biomedical Engineering, Washington University, St. Louis, Missouri 63130, USA

(Received I August 1976)

Several investigators have reported stress analyses of human aortic valves, basing their calculations on simplifying assumptions about the valve geometry. Paper’ assumed that the surface was a paraboloid of revolution, while Chong et al.* referred the leaflet geometry to two principal radii of curvature. In contrast to these approaches, Cataloglu3 and Gould et d4 used actual geometric data produced by close- range photogrammetric study of moulds made from human aortic valves. The quantitative description of the valve geometry thus obtained was then manipulated into a form suitable for input into a thin shell theory finite element computer program, and the stress distribution computed.

The research described in the present note represents an extension of the study reported by Cataloglu3 and also provides details and data not given in Gould et a1.4. It is shown that the geometric data interpretation techniques used previously3 may be replaced by a much more convenient technique based on the method of least squares. The results given in the present paper also demonstrate that an accurate stress analysis may be obtained by use of a smaller number of nodes and elements than was previously reported. In addition, stress resultants are given for three pressure loadings not considered by Cataloglu3; this completes the photogrammetric-finite element analysis of a single aortic valve under the entire pressure range experienced by a normal valve.

This work constitutes part of an on-going interdisciplinary project related to the design and development of a new prosthetic tri-leaflet aortic valve. The project aims to produce a central flow prosthetic device which closely resembles the shape and desirable hydrodynamic characteristics of the natural valve. Since the prosthetic valve should have very thin leaflets of extreme flexibility (as does the natural valve), a knowledge of the expected stress levels is thought to be an important factor in arriving at a design of sufficient strength.

Geometry and description of valve

The aortic valve is located between the left ventricle of the heart and the ascending aorta and acts as a simple check valve, permitting uni-directional flow from the ventricle to the aorta. The valve is composed of three crescent-shaped leaflets which open toward the wall of the aorta during ventricular ejection (Figure I(a)), and which close during ventricular relaxation (Figure I(b)) by sealing together.

The procedure for obtaining a numerical descriptior of the actual valve geometry has been given elsewhere5.6 and will be described here only briefly. A freshly excised human aorta was injected with silicone rubber to make six moulds under pressures of 20, 40. 60, 80, 100 and 120mmHg, respectively. A photogrammetric study of each mould was then performed, which yielded coordinates of approximately 1000 points distributed over each leaflet, the annulus. and the aortic wall. The coordinate data were

a I I I

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\’

0

Q Side view

Top view

Open

Figure 1 Schematic views of aortic valve

,’ -\

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Closed

218 Appl. Math. Modelling, 1977, Vol 1, March

Figure 2 Projections of leaflets

produced in the form of a deck of punched computer cards, to eliminate copying errors and facilitate subsequent computations. Using the photogrammetric data, the projections of the leaflets on a plane normal to the centre-line of the aorta were constructed and subsequently utilized in formulating the finite element mesh; a typical projection is shown in Figure 2 and a mesh for a single leaflet is shown in Figure 3.

Stress analysis

Assuming boundary conditions of simple support along the annular edge and a free boundary along the coapting edge, the stresses in an individual leaflet were calculated through the use of a thin shell, finite element computer program. The program was developed by Dhatt7, but subsequently modified to accept several differing forms of input data which describe the shell geometry. Triangular, shallow shell elements are used which employ a quadratic surface to represent the element geometry. The membrane and normal displacements are represented by complete cubic polynomials and a discrete Kirchoff hypothesis is used to reduce the number of degrees of freedom to nine per node. The program contains the capability for both linear and non-linear (large deformation) analyses, but several comparison test? have indicated that a linear analysis was adequate for design purposes.

To utilize the finite element program for the analysis of the valve leaflets, it was necessary to convert the coordinate data obtained in the photogrammetric study into a form compatible with the input data requirements of the program. Several different approaches are possible. Initially3, a trial and error iterative procedure was used, in which an approximating function was constructed, to interpolate the data at a small number of points on an individual valve leaflet. The difference between the approximating

Research Note

function and the original coordinate data for the leaflet was calculated at points other than the interpolating points, and if the error was found to be unacceptably large, a new set of interpolating points was chosen and the entire process was repeated. Both a single approximating function for the entire leaflet and a piecewize (patch function) approximation were used. While producing acceptable results, these procedures were time-consuming and dependent on the skill and experience of the analyst. In an effort to overcome these difficulties, a new procedure was developed. The well- known least squares curve fitting technique for a function of a single variable was generalized into a surface-fitting technique for a function of two variables.

Figure 3 Finite element mesh for single leaflet

Appl. Math. Modelling, 1977, Vol 1, March 219

Research Note

More specifically, the technique consists of minimizing the sum of squares of the deviations between real coordinates and predicted coordinates, i.e. minimize

where (x,,y,, z,,) are the coordinates of the nth point obtained from the photogrammetric analysis and

P(x,.Y,ai,%,... uk) represents a polynomial in x and y, whose coefficients a,, u2,. . , uk are to be determined from the necessary conditions for a minimum. Using a complete cubic polynomial,

~(x,.Y,~,,%~...,~,cJ =

a, + uzx + u3L’ + U&X2 + ‘.’ + a,,$

and approximately 100-200 points for an individual leaflet surface, it was found that the error between the predicted coordinate value and the original coordinate data was less than 2% for most cases; only in a few instances was the error as large as 5%. This was considered sufficiently accurate for a stress analysis to be performed. With a convenient representation of the leaflet surface now available, a finite element mesh could be constructed.

Discussion of results

In Table I, maximum stress resultants at interior points are given for pressures of 20, 100 and 120 mmHg along with results previously obtained3, for purposes of comparison. Stress resultants calculated at boundary points were also found to be reasonable; however, these points were excluded from the comparison in Table 1 due to uncertainty about the accuracy of the finite element model at boundary points. In interpreting these results, the effect of the

Table 1 Maximum stress resultants

Leaflet

1

3

Pressure

(mmHg)

20 40 60

80 100 120

20 40 60 a0

100 120

20 40 60 80

100 120

Maximum stress resultant

(g w/mm)

9.31 16.27’ 15.43’ 2 different 12.60’ meshes used 14.04‘ 10.45 lo.18

9.26 9.88’ 9.23’

10.61 l

14.79 9.05

a.23 11.72”

9.84” 9.80’

10.27 11.45

* Results by CatalogIt?

different pressures acting on the leaflet is taken into account solely through the changes in the measured leaflet shape; to facilitate comparison, a unit pressure was used as input to the finite element program in all cases. It can be seen that the maximum stress resultants obtained by the present approach are in reasonable agreement with those previously reported and in general may be considered suitable for design purposes. Since computer execution times depended strongly on the degree of mesh refinement, attention was given to the selection of the coarsest mesh which gave acceptable results. It was found that a mesh of as few as 26 nodes and 35 elements (Figure 3) gave results comparable to those obtained with two or three times as many nodes and elements. This represents a significant saving over the nearly 70 nodes and 100 elements reported by Cataloglu3. In the present study, however, attempts to use a mesh coarser than the 26 node mesh led to significant errors (i.e., results differing widely and erratically from those obtained with several more refined meshes).

Conclusions

The present research indicates that the finite element stress analysis of a human heart valve based on the data obtained by photogrammetric study yields information that is of value in designing a prosthetic device. Also the least squares surface-fitting technique has been shown to provide a more convenient and effective way to produce program input than was used previously. Finally, it appears that reasonable results can be obtained by using about 30 nodal points and approximately 40 triangular elements. Although further reduction has been attempted, it was found that too few points resulted in very large compressive stress resultants on the boundary-a condition unlikely to occur in a shell with positive Gaussian curvature and loaded in the outward normal direction.

Acknowledgement

The support of the National Institute of Health, National Heart and Lung Institute under Grant No. HL 13803 is gratefully acknowledged.

References

Poper, P. ‘Some engineering considerations in the design of prosthetic heart valves’, Progress Report, MIT, Cambridge, Mass., 1963 Chong, K. P., Hwang, N. H. C. and Wieting, D. W. 24th ACEMB, Las Vegas, 1971 Cataloglu, A. DSc Dissertation, Washington University, St. Louis, Missouri, 1974 Gould, P. L., Cataloglu, A. and Clark. R. E. Appl. Math. Modelling 1976, 1, 33 Gould, P. L., Cataloglu, A., Dhatt, G.. Chattopadhyay, A. and Clark, R. E.. J. Comput. Strut. 1973, 3, 377 Karara, S. and Marzan, G. T. Proc. 17th A. Tech. Meet. Sot. Photo-Opt. Instrum. Eng., San Diego, 1973 Dhatt, G. S. AIAAJ. 1970, p 2100 Cataloglu, A.. Gould, P. L. and Clark, R. E. J. Biomech. 1975 8,

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