Med. & Biol. Eng. & Comput., 1979, 17, 25-30
Orthotropic elastic moduli for left ventricular mechanical behaviour A. L. Y e t t r a m
C . A . Vinson
Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, England
A b s t r a c t - - T h e determination of numerical values for the elasticity of the myocardium is a prerequisite for any accurate analysis of the left ventricle. Such values as are normally quoted are for a modulus of elasticity for an isotropic homogeneous m o d e l Here some current suggestions for the modulus of elasticity are collated and a further analysis for values on the assumption of an orthotropic left ventricle is presented. K e y w o r d s - - E l a s t i c modulus, Left ventricle, Orthotropy
1 Introduction
ATTEMPTING to perform a mechanical and structural analysis of the human left ventricle has recently occupied much research effort. The problem from an engineering standpoint is extremely intricate if it is to be undertaken completely and with as much accuracy as possible. The two main areas where difficulty arises are concerned with obtaining geometrical and material property data. Considering the material properties of the myocardium, these are likely to be nonlinear, viscoelastic, anisotropic and nonhomogeneous during the diastolic phase. In the systolic phase this complexity is further compounded by selfenergisation. Many investigators have attempted to examine what they have termed the elasticity of the heart. A large number of these have been based on values of dP/dV where V (volume) is determined from a spherical or ellipsoidal assumption. Nonlinear property values have also been derived using an exponential relationship between P (pressure) and 1,1. Examples of such relationships have been reviewed by MIRSKY (1976) and GmSON (1977). It is not reasonable to quantify an engineering definition of muscle stiffness from analyses of P-V recordings that can give no indication of local property values. Also, they cannot differentiate between a shape change and a property change. Unfortunately, values for a comprehensive engineering description of the myocardium are not, as yet, known. In engineering terms, the values most frequently suggested are those for the modulus of elasticity (E), which assumes small deflection,
linear elasticity and homogeneity and also the tangent modulus (k) which allows for a nonlinear material by including the rate of change of stress to that of strain. Large deformation analyses have been carried out by MIRSKY (1973) and DEMIRAY (1976), who both assumed the ventricle to be spherical, and by JANZ and GRIMM (1974), who used finite element techniques with a model whose geometry was that of a truncated ellipsoid. There are a few suggestions for a comprehensive constitutive equation for the myocardium in the literature (PINTO and FUNG, 1973; HUNTER, 1975).
12 1( _.
f.~...~./'"-"~.~.~
~.
~
9
9
o
4'0 point
'
number
Fig. I The inability of an isotropic model to match the true long axis length of the left ventricle ....... . . . . . . . . . . .
First received 12th January and in final form 22nd March 1978
minor axis from echocardiography long axis from cineangiocardiography long axis predicted by isotropic ellipsoidal model
0140-0118/79/010025 4- 06 $01.50/0 @ IFMBE: 1979
Medical & Biological Engineering & Computing D
January1979
25
The values for these parameters may be found by either purely experimental testing or from observation of deflections used in conjunction with measured forces to calculate the material properties using an analogous model. Ideally, for a complete modelling of the left ventricle the use of a 3-dimensional reconstruction of the left ventricle by finite elements is indicated. However, there is a lack of data concerning 3-dimensional anisotropic properties. To establish a starting point for the material properties to be used later in such a finite element analysis, here the equations for an orthotropic ellipsoidal shell are examined. These are reversed to calculate the two moduli of elasticity incrementally throughout the cycle from measured dimension, long axis length, wall thickness and intraventricular pressure.
2 Derivation from mechanical testing F o r any material, properties in three dimensions are difficult to obtain. In the case of the myocardium it is further complicated by the ethics in carrying out experimental tests on humans and the complex movement and geometry of the heart in vivo. It is, therefore, only possible to test 1-dimensional excised strips of cadaveric cardiac material. These strips may be preserved, in the hopefully short time between excision and testing, in a suitable solution. They will nevertheless be reacting in a different environment to that of the heart in vivo. In WOHLISH and CLAMANN (1936) tests were reported on strips of frog ventricle. Their passive 1-dimensional tests gave a modulus of elasticity (E) of 18 g/cm 2 (1.76 k N / m 2) to 440 g/cm 2 (43"2 k N / m2). LUNDIN (1944) tested ventricular muscle and found E to be in the range 100x 10 a dyne/era 2 to 1 0 0 0 x l 0 3 dyne/cm 2 ( 1 0 - 1 0 0 k N / m 2) in passive strips. When the muscular fibres were contracted isometrically, the value of E ranged from 800 x 103 dyne/cm 2 to 1 0 0 0 0 x l 0 3 dyne/cm 2 (80-1000 kN/m2). JANZ and GRIMM (1972) provide a more comprehensive orthotropic set of experimental values for rat cardiac muscle. In general, these results can be compared with the others by taking their suggested value for an isotropic outer twothirds of the ventricle of 60 g / c m 2 (5-88 kN/m2). W o r k also continues in trying to find an overall constitutive equation for the muscle, that fully describes it in all its complexity. Such equations can be found in the work of PINTO (1973), and, more recently, from I-IuNxER (1975). However, the relevance of properties derived from in vitro 1-dimensional strips to the 3-dimensional in vivo ventricle is questionable. The proof of such values will presumably lie in combining them into an engineering finite element analysis, with accurate geometry, reproducing the movement of the ventricle throughout the cycle in normal and abnormal conditions and then comparing the predicted and observed results. 26
3 Derivation from geometric models The material properties used in an engineering analysis are defined by a stress/strain relationship. Some investigators have derived stress/strain relationships for the left ventricle by taking observed deflections in vivo and comparing them with the movement of an analogous model. FRONEK et al. (1967) assumed a circular midsection for a canine left ventricle and their observations suggested a diastolic value of E of 77x103 dyne/cm 2 (7.7 kN/m2). GmSTA and SANDLER (1969) proposed, from the observed deflections of their ellipsoidal model, that E in diastole was 50x 103 dyne/cm 2 (5 kN/m2). Also in that year, SANDLER and GHISTA (1969) published another paper where they calculated a diastolic E of 6-10 k N / m 2 and a systolic E of 40-70 k N / m 2, once again based on an ellipsoidal model. LAFFERTYet al. (1972) used a spherical model and suggested an end-systolic modulus of 900x 103 dyne/era 2 (90kN/m2). GIBSON and BROWN (1974) used an ellipsoidal Laplace-based model and
12
~120 ~
E
8C
~ 40 ~.
ot c
/ \'-.........~"
8o0
400 ~fi 0 ~, -~ t~~ -400
40 point number
_A 120
-800 Fig. 2 Eo calculated from orthotropic ellipsoidal model (patient with mitral regurgitation) with geometric and pressure data above
Eo . . . . . ..... . . . . . .
long axis dimension pressure
Time for one cardiac cycle 0 . 7 6 9 s points on abcissa
Medical & Biological Engineering & Computing
or 129
January 1979
published data relating stress to strain throughout the cardiac cycle. Also in that year~ MIRSKY and PARMLEY (1974) published values for end diastole of 37-186 k N / m z, these values having been found using an ellipsoidal model with a specified exponential P V relationship. Their work also suggests values for a tangent modulus. GHISTA et al. (1975) developed the earlier method of GHISTA and SANDLER (1969) using the same ellipsoidal thick shell model to predict a homogeneous isotropic linearly-elastic value of E throughout the cycle. Values of E were found for diastole ranging from 7 - 1 1 0 k N / m 2 and for systole from 27-400 k N / m 2.
4 Extension to derive orthotropie properties An ellipsoid, based on membrane theory and linear elasticity, provides the basic model. Here, however, the possibility of the ellipsoid being orthotropic, i.e. with an elastic modulus E , in the longitudinal or meridional direction being different from that, Eo, in the latitudinal or circumferential direction, has been incorporated. The geometry was supplied by GIBSON and BRown* in the form of short axis echocardiographic and long axis cineangiocardiographic recordings. Corresponding pressure values, which had been measured by a pressure tipped catheter, were also provided. The derivation of the elasticity equations for an orthotropic ellipsoidal shell have been presented by VINSON (1977) and a summary of the relevant relationships are shown in the Appendix. The cycle was broken into 128 incremental steps, irrespective of the cycle length. The time axis is therefore nondimensionalised. The long axis data were found from cineangiocardiograms and minor axis data and wall thicknesses from echocardiography. The ventricular wall thickness rather than the septum thickness was used to specify the wall thickness of the mathematical model. As only an average of twenty cineangiocardiographic frames were available for each cycle, a polynomial was produced to describe long axis length at all the 128 steps taken in the cycle. Analysis with the particular case of E , = Eo, the isotropic model, was carried out. F o r this case, considering each incremental step, experimental data are available that may be used in the two simultaneous equations, if Poisson's ratio is prescribed, to calculate E and long axis length. However, the isotropic model showed that it was unable to match the long axis movement of the left ventricle. Fig. 1 shows the difference between the estimated long axis length from cineangiocardiograms and that predicted by this mathematical model. The long axis length for the model is seen to diverge "D. G. GIBSON and D, J. BROWN. London, England
The Bfompton Hospital.
Medical & Biological Engineering & Computing
rapidly from reality. This would produce a spherical shape by the long axis nearly equalling the equatorial diameter, the 'dimension' found by echocardiography. This feature has implications in examining any isotropic work, such as that by GmsxA (1975), and dearly shows that when ellipsoidal models are used the assumptions of isotropy produces grossly inaccurate results. Results from the orthotropic analysis are illustrated in Figs. 2 and 3. Here the analysis uses the observed data for long axis length, dimension, wall thickness and pressure and their incremental changes; /t,o is prescribed, this allows E,, Eo and /tom to be calculated from eqns. 1, 2 and 3. These graphs also show the pressure and major and minor axis changes throughout the cycle. Only E0 is shown as Eo/E,~ was found to be approximately in the ratio 1:1"1. In fact, when the orthotropic results are examined, they show that small changes in the ratio E,/Eo, for these ellipsoidal models, can accommodate the proper long axis change. The results suggest values of Eo for diastole of 0-100 k N / m 2 and a peak in systole of over 1000
/./"~
~120 ~'80
~2
.~---~"
4O
2
i
. . . . . . . . . . . .
E u
-C
4~ Q
80C
40C
:.of2
120
point number
-400
-800
Fig. 3 Eo calculated
from orthotropic e//ipsoidal model (patient with hypertrophic cardiomyopath)/) with geometric and pressure data above Eo long axis dimension pressure Time for one cardiac cycle 0 . 8 4 6 s or 129 points on abcissa
January 1979
27
k N / m 2. The graphs show a wide variation between the patients and through each phase of the cycle for any one analysis. As the equations are for a passive ellipsoidal structure under certain combinations of movement and change in pressure in the left ventricle, negative values of E are predicted by the equations. These negative values would imply muscular activity r a t h e r than passivity and are indeed found in systole and at start diastole before minimum pressure. Using an ellipsoidal model to calculate mechanical properties employs obvious assumptions in the defined shape that reduce the degrees of freedom so that only a restricted number of material constants may be found. As the use of the finite element method is a logical step in refining a stress and deflection analysis for the left ventricle, then so also might it provide the basis for the derivation of properties. RAy et al. (1976) used finite elements to optimise the motion observed in diastole and calculated a value of 50 k N / m 2. PAO et aL (1975) used a simpler cross-sectional analysis and suggested a diastolic value of 92 k N / m 2. NIKRAVESH (1976) derived values for E by using an optimisation of the output deflections from a finite element model of the left ventricle based on single plane angiocardiographs. The model was considered to be isotropic, linearly elastic and to be built from homogeneous layers (rings formed by brick elements). The value of E of each ring was
Wr
calculated. He found that the model predicted a gradual fall in E from 5 • 104 dyne/cm 2 (5 k N / m 2) at the base to 0 . 5 x 104 dyne/cm z (0"5 k N / m 2) at the apex. However, all the above analyses have some simplifying assumption concerning the 3dimensional geometry. Thus, extension of the use of finite-element models to help predict material properties in a possibly anisotropic and nonhomogeneous myocardium appears to be indicated. A basic technique has been suggested by KAVANAGH (1973). Unfortunately, however, the equations which govern the values of material properties in terms of observed forces and deflections are ill conditioned. An attempt has been made to use the finite element method to predict nonhomogenous properties in this way (i.e. in the reverse manner to a normal finite-element analysis) but it has been found that the input data required need to be of such high quality and the computing time so large as to make this approach impracticable, at least at the present time (VINSON, 1977). The myocardium is generally regarded as being incompressible, or close to that state, thus a Poisson's ratio close to 0.5 is appropriate. HAMID and GnISrA (1974) and NIKRAVESH(1976) have used 0" 45 in their work, while JANZ and GRIMM (1972) have used 0" 49. When considering equations that are anisotropic in three dimensions, then six Poisson's ratios are required. It is likely that some of these may be allowed to differ much more from 0" 5 to accommodate the anisotropy while presumably still maintaining near incompressibility. 5 Conclusions
I
b_
W~, :90 I-
-l
Fig. 4 Ellipsoidal shell geometry 28
The stiffness of the myocardium is extremely complex and is undoubtedly nonhomogeneous, nonlinear, viscoelastic and anisotropic. An analysis to evaluate orthotropic parameters is presented here, where the cycle is divided into a series of small increments each approximating to a linearly elastic step. The results found by this method are however constrained by the ellipsoidal s h a p e and membrane theory assumptions used. F r o m the few patients who have been evaluated, it can be said that the values of E throughout the cycle vary greatly, even when the aetiology is similar. Values of Eo, on average, for the patients studied were 4 7 k N / m z in diastole and 1 2 7 5 k N / m z in systole. However, as the values within diastole and systole also show a large variation, they cannot be clinically useful in themselves. The most significant difference found was between patients with regurgitation (Fig. 2) and a patient with cardiomyopathy (Fig. 3). The latter ventricle appeared to be very flexible in diastole, yet no great change in dimension occurred, and also very flexible in systole where a normal, rather than as might have been expected a reduced, pressure arose. Thus, care must be taken in
Medical & Biological Engineering & Computing
January 1979
LUNDIN, G. (1944) Mechanical properties of cardiac muscle. Acta Physiologica Scandinavica 7, suppl. XX, 80. MIRSKY,I. (1973) Ventricular and arterial wall stresses-based in large deformation analyses. Biophysical J. 13, 1141. MIRSKY, I. and PArtMLEV,W. W. (1974) Evaluation of passive elastic stiffness for the left ventricle and isolated heart muscle. In Cardiac mechanics, John Wiley, New York. MJRSKY, I. (1976) Assessment of passive elastic stiffness of cardiac muscle: Mathematical concepts, physiologic and clinical considerations, directions of future research. Progress in Cardiovascular Diseases 18, 277. NIKr~v~sn, E. P.. (1976) Optimisation in finite element analysis with special reference to three-dimensional left ventricular dynamics. D.Eng. thesis, Department of Electrical Engineering, Tulane University, New Orleans, Louisiana. PAO, Y. C., RITMAN~E. L., RoBs, R. A. and WOOD, E. H. (1975) A finite element method for evaluating cross-sectional Young's Modulus of diastolic left References ventricle. Proceedings of 28th meeting of ACEMB, 385. l~wro, J. and FUNG,Y. C. (1973) Mechanical properties DEMIRAY, H. (1976) Stresses in the ventricular wall. of the heart muscle. 1973 biomechanics symposium Trans. A S M E , 194. ASME, AMD 2, 37. FRONEK, A., COX, R., MvRoo, J. P. and PETERSON, RAY, G., CHANDRAN,K. B., NIKRAVESH,E. P., GI-IISTA, L. H. (1976) ln-vivo elastic modulus of cardiac muscle. D. N. and SANDLER,H. (1976) Estimation of the local Proceedings of the 50th annual meeting of the Federaelastic modulus of the normal and infarcted left tion of American Societies for experimental biology ventricle from angiocardiographic data. Proceedings 26, 382. of the 4th New England bioengineering conference, 173. GmswA, D. N. and SANDLER, H. (1969) An analytic elastic-viscoelastic model for the shape and forces in SANDLER, H. and GmSTA,D. N. (1969) Mechanical and the left ventricle. J. Biomech. 2, 35. dynamic implications of dimensional measurements of the left ventricle. Proceedings of the 52rid annual GHISTA,D. N., SANDLER,H. and VAvo, W. H. (1975) meeting of the Federation of American Societies for Elastic modulus of the human intact left ventricle-experimental biology 28, 1344. determination and physiological interpretations. Med. & Biol. Eng. 13, 151. VINSON, C. A. (1977) Analysis of stress and deformation in the human left ventricle. Ph.D. thesis, Department GIBSON,D. G. and BROWN,D. J. (1974) Relation between of Mechanical Engineering, Brunel University, Uxdiastolic left ventricular wall stress and strain in man. bridge, Middlesex, England. Brit. Heart d. 36, 1066. WOHLXSCH, E. and CL~a~AN,H. G. (1936) Untersuchungen GmSON,D. G. (1977) Clinical assessment of left ventricuiiber das elastische verhalten des ruhende herzmuskels. lar function. In Recent advances in cardiology (Ed. J. Plugers Arch. 590. Hamer), Churchill Livingstone. I-[AMID, i . S. and GHISTA,D. N. (1974) Finite element analysis of cardiac structures. In Finite element methods in engineering, University of New South Wales. HUNTER, P. J. (1975) Finite element analysis of cardiac muscle mechanics. D.Phil. thesis, University of Oxford. JANZ, R. F. and GRIMM, A. F. (1972) Finite element model for the mechanical behaviour of the left ventricle. Circ. Res. 30, 244. JANZ, R. F., KUBERT,B. R. and MOmARTY,T. F. (1974) Deformation of the diastolic left ventricle. II Non- Appendix linear geometric effects. J. Biomeeh. 7, 509. KAVANAGH, K. T. (1973) Experiment versus analysis: Solution for the orthotropic case Computational techniques for the description of static material response. Int. J. Num. Meths. Eng. 5, 503. The model is considered to have four variable elastic LAFFERTY,J. F., McCuTcHEON, E. P., FUNK, J. E. and properties: circumferential and longitudinal moduli of HIGGINS,A. M. (1972) In vivo determination of elastic elasticity Eo and E,, and two corresponding Poisson's modulus of canine cardiac muscle. J. Basic Eng. 94, ratios p-,e and ~0,. The directions of displacement are 912. illustrated in Fig. 4.
the use of what are normally models for passive engineering analysis. Certainly, one might expect a flexible ventricle to allow greater filling in diastole and also reduce ejection, but these models are for passive equilibrium, not for selfactivation. The value of Eo/Eo suggested is important as an indicator of anisotropy. A finite-element stress analysis (VINsON, 1977) did not corroborate the suggested ratio found in the ellipsoidal analysis. In that case, the movement presumably causing the orthotropic value was accommodated by localised deformation. This highlights the fact that both true geometry and material properties are necessary. Hence simplified ellipsoidaI shapes may mask the significant part of the left ventricular movement when they are used to model the ventricle for both material property determination and also for stress and deformation analysis.
Medical & Biological Engineering & Computing
January 1979
29
In the above
The solution for 17o is given as
pa 2
az
~/zF I1
Owing to the restricted number of unknowns to form suitable equations, it is necessary to specify one variable. Taking this to be t~0 and using the calculated value of Eo, then the remaining two unknowns can be found as
(a z sin 2 $ + b 2 cos ~ 4,)2 sin r
oJ (a 2 sin z ~ + b 2 cos z ~b)sin
a~
(1 - ~,~o) /
(b 1 1,, t[;{WT-~42h
E. =
lz
J
d4,
E--~ ,
13 =
sin ~b
(4)
(5)
(6)
o
-- {~z --2be F~,olx--2Iz(1--Fc, o)} Eo]
(2)
and Foo ~ ~
30
IJ'r
.
.
.
.
.
.
.
.
.
.
(3)
and P is the increment of applied pressure, a is the major axis half-length (dimension), b is the minor axis halflength (long axis), h is the wall thickness, W~=o is the incremental change in b, W, =90 is the incremental change in a, Eo, E~, are the two moduli of elasticity in the circumferential and longitudinal directions and t~,0, t~0, are the two Poisson's ratios.
Medical & Biological Engineering & Computing
January 1979
Orthotropic elastic moduli for left ventricular mechanical behaviour A. L. Y e t t r a m
C . A . Vinson
Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, England
A b s t r a c t - - T h e determination of numerical values for the elasticity of the myocardium is a prerequisite for any accurate analysis of the left ventricle. Such values as are normally quoted are for a modulus of elasticity for an isotropic homogeneous m o d e l Here some current suggestions for the modulus of elasticity are collated and a further analysis for values on the assumption of an orthotropic left ventricle is presented. K e y w o r d s - - E l a s t i c modulus, Left ventricle, Orthotropy
1 Introduction
ATTEMPTING to perform a mechanical and structural analysis of the human left ventricle has recently occupied much research effort. The problem from an engineering standpoint is extremely intricate if it is to be undertaken completely and with as much accuracy as possible. The two main areas where difficulty arises are concerned with obtaining geometrical and material property data. Considering the material properties of the myocardium, these are likely to be nonlinear, viscoelastic, anisotropic and nonhomogeneous during the diastolic phase. In the systolic phase this complexity is further compounded by selfenergisation. Many investigators have attempted to examine what they have termed the elasticity of the heart. A large number of these have been based on values of dP/dV where V (volume) is determined from a spherical or ellipsoidal assumption. Nonlinear property values have also been derived using an exponential relationship between P (pressure) and 1,1. Examples of such relationships have been reviewed by MIRSKY (1976) and GmSON (1977). It is not reasonable to quantify an engineering definition of muscle stiffness from analyses of P-V recordings that can give no indication of local property values. Also, they cannot differentiate between a shape change and a property change. Unfortunately, values for a comprehensive engineering description of the myocardium are not, as yet, known. In engineering terms, the values most frequently suggested are those for the modulus of elasticity (E), which assumes small deflection,
linear elasticity and homogeneity and also the tangent modulus (k) which allows for a nonlinear material by including the rate of change of stress to that of strain. Large deformation analyses have been carried out by MIRSKY (1973) and DEMIRAY (1976), who both assumed the ventricle to be spherical, and by JANZ and GRIMM (1974), who used finite element techniques with a model whose geometry was that of a truncated ellipsoid. There are a few suggestions for a comprehensive constitutive equation for the myocardium in the literature (PINTO and FUNG, 1973; HUNTER, 1975).
12 1( _.
f.~...~./'"-"~.~.~
~.
~
9
9
o
4'0 point
'
number
Fig. I The inability of an isotropic model to match the true long axis length of the left ventricle ....... . . . . . . . . . . .
First received 12th January and in final form 22nd March 1978
minor axis from echocardiography long axis from cineangiocardiography long axis predicted by isotropic ellipsoidal model
0140-0118/79/010025 4- 06 $01.50/0 @ IFMBE: 1979
Medical & Biological Engineering & Computing D
January1979
25
The values for these parameters may be found by either purely experimental testing or from observation of deflections used in conjunction with measured forces to calculate the material properties using an analogous model. Ideally, for a complete modelling of the left ventricle the use of a 3-dimensional reconstruction of the left ventricle by finite elements is indicated. However, there is a lack of data concerning 3-dimensional anisotropic properties. To establish a starting point for the material properties to be used later in such a finite element analysis, here the equations for an orthotropic ellipsoidal shell are examined. These are reversed to calculate the two moduli of elasticity incrementally throughout the cycle from measured dimension, long axis length, wall thickness and intraventricular pressure.
2 Derivation from mechanical testing F o r any material, properties in three dimensions are difficult to obtain. In the case of the myocardium it is further complicated by the ethics in carrying out experimental tests on humans and the complex movement and geometry of the heart in vivo. It is, therefore, only possible to test 1-dimensional excised strips of cadaveric cardiac material. These strips may be preserved, in the hopefully short time between excision and testing, in a suitable solution. They will nevertheless be reacting in a different environment to that of the heart in vivo. In WOHLISH and CLAMANN (1936) tests were reported on strips of frog ventricle. Their passive 1-dimensional tests gave a modulus of elasticity (E) of 18 g/cm 2 (1.76 k N / m 2) to 440 g/cm 2 (43"2 k N / m2). LUNDIN (1944) tested ventricular muscle and found E to be in the range 100x 10 a dyne/era 2 to 1 0 0 0 x l 0 3 dyne/cm 2 ( 1 0 - 1 0 0 k N / m 2) in passive strips. When the muscular fibres were contracted isometrically, the value of E ranged from 800 x 103 dyne/cm 2 to 1 0 0 0 0 x l 0 3 dyne/cm 2 (80-1000 kN/m2). JANZ and GRIMM (1972) provide a more comprehensive orthotropic set of experimental values for rat cardiac muscle. In general, these results can be compared with the others by taking their suggested value for an isotropic outer twothirds of the ventricle of 60 g / c m 2 (5-88 kN/m2). W o r k also continues in trying to find an overall constitutive equation for the muscle, that fully describes it in all its complexity. Such equations can be found in the work of PINTO (1973), and, more recently, from I-IuNxER (1975). However, the relevance of properties derived from in vitro 1-dimensional strips to the 3-dimensional in vivo ventricle is questionable. The proof of such values will presumably lie in combining them into an engineering finite element analysis, with accurate geometry, reproducing the movement of the ventricle throughout the cycle in normal and abnormal conditions and then comparing the predicted and observed results. 26
3 Derivation from geometric models The material properties used in an engineering analysis are defined by a stress/strain relationship. Some investigators have derived stress/strain relationships for the left ventricle by taking observed deflections in vivo and comparing them with the movement of an analogous model. FRONEK et al. (1967) assumed a circular midsection for a canine left ventricle and their observations suggested a diastolic value of E of 77x103 dyne/cm 2 (7.7 kN/m2). GmSTA and SANDLER (1969) proposed, from the observed deflections of their ellipsoidal model, that E in diastole was 50x 103 dyne/cm 2 (5 kN/m2). Also in that year, SANDLER and GHISTA (1969) published another paper where they calculated a diastolic E of 6-10 k N / m 2 and a systolic E of 40-70 k N / m 2, once again based on an ellipsoidal model. LAFFERTYet al. (1972) used a spherical model and suggested an end-systolic modulus of 900x 103 dyne/era 2 (90kN/m2). GIBSON and BROWN (1974) used an ellipsoidal Laplace-based model and
12
~120 ~
E
8C
~ 40 ~.
ot c
/ \'-.........~"
8o0
400 ~fi 0 ~, -~ t~~ -400
40 point number
_A 120
-800 Fig. 2 Eo calculated from orthotropic ellipsoidal model (patient with mitral regurgitation) with geometric and pressure data above
Eo . . . . . ..... . . . . . .
long axis dimension pressure
Time for one cardiac cycle 0 . 7 6 9 s points on abcissa
Medical & Biological Engineering & Computing
or 129
January 1979
published data relating stress to strain throughout the cardiac cycle. Also in that year~ MIRSKY and PARMLEY (1974) published values for end diastole of 37-186 k N / m z, these values having been found using an ellipsoidal model with a specified exponential P V relationship. Their work also suggests values for a tangent modulus. GHISTA et al. (1975) developed the earlier method of GHISTA and SANDLER (1969) using the same ellipsoidal thick shell model to predict a homogeneous isotropic linearly-elastic value of E throughout the cycle. Values of E were found for diastole ranging from 7 - 1 1 0 k N / m 2 and for systole from 27-400 k N / m 2.
4 Extension to derive orthotropie properties An ellipsoid, based on membrane theory and linear elasticity, provides the basic model. Here, however, the possibility of the ellipsoid being orthotropic, i.e. with an elastic modulus E , in the longitudinal or meridional direction being different from that, Eo, in the latitudinal or circumferential direction, has been incorporated. The geometry was supplied by GIBSON and BRown* in the form of short axis echocardiographic and long axis cineangiocardiographic recordings. Corresponding pressure values, which had been measured by a pressure tipped catheter, were also provided. The derivation of the elasticity equations for an orthotropic ellipsoidal shell have been presented by VINSON (1977) and a summary of the relevant relationships are shown in the Appendix. The cycle was broken into 128 incremental steps, irrespective of the cycle length. The time axis is therefore nondimensionalised. The long axis data were found from cineangiocardiograms and minor axis data and wall thicknesses from echocardiography. The ventricular wall thickness rather than the septum thickness was used to specify the wall thickness of the mathematical model. As only an average of twenty cineangiocardiographic frames were available for each cycle, a polynomial was produced to describe long axis length at all the 128 steps taken in the cycle. Analysis with the particular case of E , = Eo, the isotropic model, was carried out. F o r this case, considering each incremental step, experimental data are available that may be used in the two simultaneous equations, if Poisson's ratio is prescribed, to calculate E and long axis length. However, the isotropic model showed that it was unable to match the long axis movement of the left ventricle. Fig. 1 shows the difference between the estimated long axis length from cineangiocardiograms and that predicted by this mathematical model. The long axis length for the model is seen to diverge "D. G. GIBSON and D, J. BROWN. London, England
The Bfompton Hospital.
Medical & Biological Engineering & Computing
rapidly from reality. This would produce a spherical shape by the long axis nearly equalling the equatorial diameter, the 'dimension' found by echocardiography. This feature has implications in examining any isotropic work, such as that by GmsxA (1975), and dearly shows that when ellipsoidal models are used the assumptions of isotropy produces grossly inaccurate results. Results from the orthotropic analysis are illustrated in Figs. 2 and 3. Here the analysis uses the observed data for long axis length, dimension, wall thickness and pressure and their incremental changes; /t,o is prescribed, this allows E,, Eo and /tom to be calculated from eqns. 1, 2 and 3. These graphs also show the pressure and major and minor axis changes throughout the cycle. Only E0 is shown as Eo/E,~ was found to be approximately in the ratio 1:1"1. In fact, when the orthotropic results are examined, they show that small changes in the ratio E,/Eo, for these ellipsoidal models, can accommodate the proper long axis change. The results suggest values of Eo for diastole of 0-100 k N / m 2 and a peak in systole of over 1000
/./"~
~120 ~'80
~2
.~---~"
4O
2
i
. . . . . . . . . . . .
E u
-C
4~ Q
80C
40C
:.of2
120
point number
-400
-800
Fig. 3 Eo calculated
from orthotropic e//ipsoidal model (patient with hypertrophic cardiomyopath)/) with geometric and pressure data above Eo long axis dimension pressure Time for one cardiac cycle 0 . 8 4 6 s or 129 points on abcissa
January 1979
27
k N / m 2. The graphs show a wide variation between the patients and through each phase of the cycle for any one analysis. As the equations are for a passive ellipsoidal structure under certain combinations of movement and change in pressure in the left ventricle, negative values of E are predicted by the equations. These negative values would imply muscular activity r a t h e r than passivity and are indeed found in systole and at start diastole before minimum pressure. Using an ellipsoidal model to calculate mechanical properties employs obvious assumptions in the defined shape that reduce the degrees of freedom so that only a restricted number of material constants may be found. As the use of the finite element method is a logical step in refining a stress and deflection analysis for the left ventricle, then so also might it provide the basis for the derivation of properties. RAy et al. (1976) used finite elements to optimise the motion observed in diastole and calculated a value of 50 k N / m 2. PAO et aL (1975) used a simpler cross-sectional analysis and suggested a diastolic value of 92 k N / m 2. NIKRAVESH (1976) derived values for E by using an optimisation of the output deflections from a finite element model of the left ventricle based on single plane angiocardiographs. The model was considered to be isotropic, linearly elastic and to be built from homogeneous layers (rings formed by brick elements). The value of E of each ring was
Wr
calculated. He found that the model predicted a gradual fall in E from 5 • 104 dyne/cm 2 (5 k N / m 2) at the base to 0 . 5 x 104 dyne/cm z (0"5 k N / m 2) at the apex. However, all the above analyses have some simplifying assumption concerning the 3dimensional geometry. Thus, extension of the use of finite-element models to help predict material properties in a possibly anisotropic and nonhomogeneous myocardium appears to be indicated. A basic technique has been suggested by KAVANAGH (1973). Unfortunately, however, the equations which govern the values of material properties in terms of observed forces and deflections are ill conditioned. An attempt has been made to use the finite element method to predict nonhomogenous properties in this way (i.e. in the reverse manner to a normal finite-element analysis) but it has been found that the input data required need to be of such high quality and the computing time so large as to make this approach impracticable, at least at the present time (VINSON, 1977). The myocardium is generally regarded as being incompressible, or close to that state, thus a Poisson's ratio close to 0.5 is appropriate. HAMID and GnISrA (1974) and NIKRAVESH(1976) have used 0" 45 in their work, while JANZ and GRIMM (1972) have used 0" 49. When considering equations that are anisotropic in three dimensions, then six Poisson's ratios are required. It is likely that some of these may be allowed to differ much more from 0" 5 to accommodate the anisotropy while presumably still maintaining near incompressibility. 5 Conclusions
I
b_
W~, :90 I-
-l
Fig. 4 Ellipsoidal shell geometry 28
The stiffness of the myocardium is extremely complex and is undoubtedly nonhomogeneous, nonlinear, viscoelastic and anisotropic. An analysis to evaluate orthotropic parameters is presented here, where the cycle is divided into a series of small increments each approximating to a linearly elastic step. The results found by this method are however constrained by the ellipsoidal s h a p e and membrane theory assumptions used. F r o m the few patients who have been evaluated, it can be said that the values of E throughout the cycle vary greatly, even when the aetiology is similar. Values of Eo, on average, for the patients studied were 4 7 k N / m z in diastole and 1 2 7 5 k N / m z in systole. However, as the values within diastole and systole also show a large variation, they cannot be clinically useful in themselves. The most significant difference found was between patients with regurgitation (Fig. 2) and a patient with cardiomyopathy (Fig. 3). The latter ventricle appeared to be very flexible in diastole, yet no great change in dimension occurred, and also very flexible in systole where a normal, rather than as might have been expected a reduced, pressure arose. Thus, care must be taken in
Medical & Biological Engineering & Computing
January 1979
LUNDIN, G. (1944) Mechanical properties of cardiac muscle. Acta Physiologica Scandinavica 7, suppl. XX, 80. MIRSKY,I. (1973) Ventricular and arterial wall stresses-based in large deformation analyses. Biophysical J. 13, 1141. MIRSKY, I. and PArtMLEV,W. W. (1974) Evaluation of passive elastic stiffness for the left ventricle and isolated heart muscle. In Cardiac mechanics, John Wiley, New York. MJRSKY, I. (1976) Assessment of passive elastic stiffness of cardiac muscle: Mathematical concepts, physiologic and clinical considerations, directions of future research. Progress in Cardiovascular Diseases 18, 277. NIKr~v~sn, E. P.. (1976) Optimisation in finite element analysis with special reference to three-dimensional left ventricular dynamics. D.Eng. thesis, Department of Electrical Engineering, Tulane University, New Orleans, Louisiana. PAO, Y. C., RITMAN~E. L., RoBs, R. A. and WOOD, E. H. (1975) A finite element method for evaluating cross-sectional Young's Modulus of diastolic left References ventricle. Proceedings of 28th meeting of ACEMB, 385. l~wro, J. and FUNG,Y. C. (1973) Mechanical properties DEMIRAY, H. (1976) Stresses in the ventricular wall. of the heart muscle. 1973 biomechanics symposium Trans. A S M E , 194. ASME, AMD 2, 37. FRONEK, A., COX, R., MvRoo, J. P. and PETERSON, RAY, G., CHANDRAN,K. B., NIKRAVESH,E. P., GI-IISTA, L. H. (1976) ln-vivo elastic modulus of cardiac muscle. D. N. and SANDLER,H. (1976) Estimation of the local Proceedings of the 50th annual meeting of the Federaelastic modulus of the normal and infarcted left tion of American Societies for experimental biology ventricle from angiocardiographic data. Proceedings 26, 382. of the 4th New England bioengineering conference, 173. GmswA, D. N. and SANDLER, H. (1969) An analytic elastic-viscoelastic model for the shape and forces in SANDLER, H. and GmSTA,D. N. (1969) Mechanical and the left ventricle. J. Biomech. 2, 35. dynamic implications of dimensional measurements of the left ventricle. Proceedings of the 52rid annual GHISTA,D. N., SANDLER,H. and VAvo, W. H. (1975) meeting of the Federation of American Societies for Elastic modulus of the human intact left ventricle-experimental biology 28, 1344. determination and physiological interpretations. Med. & Biol. Eng. 13, 151. VINSON, C. A. (1977) Analysis of stress and deformation in the human left ventricle. Ph.D. thesis, Department GIBSON,D. G. and BROWN,D. J. (1974) Relation between of Mechanical Engineering, Brunel University, Uxdiastolic left ventricular wall stress and strain in man. bridge, Middlesex, England. Brit. Heart d. 36, 1066. WOHLXSCH, E. and CL~a~AN,H. G. (1936) Untersuchungen GmSON,D. G. (1977) Clinical assessment of left ventricuiiber das elastische verhalten des ruhende herzmuskels. lar function. In Recent advances in cardiology (Ed. J. Plugers Arch. 590. Hamer), Churchill Livingstone. I-[AMID, i . S. and GHISTA,D. N. (1974) Finite element analysis of cardiac structures. In Finite element methods in engineering, University of New South Wales. HUNTER, P. J. (1975) Finite element analysis of cardiac muscle mechanics. D.Phil. thesis, University of Oxford. JANZ, R. F. and GRIMM, A. F. (1972) Finite element model for the mechanical behaviour of the left ventricle. Circ. Res. 30, 244. JANZ, R. F., KUBERT,B. R. and MOmARTY,T. F. (1974) Deformation of the diastolic left ventricle. II Non- Appendix linear geometric effects. J. Biomeeh. 7, 509. KAVANAGH, K. T. (1973) Experiment versus analysis: Solution for the orthotropic case Computational techniques for the description of static material response. Int. J. Num. Meths. Eng. 5, 503. The model is considered to have four variable elastic LAFFERTY,J. F., McCuTcHEON, E. P., FUNK, J. E. and properties: circumferential and longitudinal moduli of HIGGINS,A. M. (1972) In vivo determination of elastic elasticity Eo and E,, and two corresponding Poisson's modulus of canine cardiac muscle. J. Basic Eng. 94, ratios p-,e and ~0,. The directions of displacement are 912. illustrated in Fig. 4.
the use of what are normally models for passive engineering analysis. Certainly, one might expect a flexible ventricle to allow greater filling in diastole and also reduce ejection, but these models are for passive equilibrium, not for selfactivation. The value of Eo/Eo suggested is important as an indicator of anisotropy. A finite-element stress analysis (VINsON, 1977) did not corroborate the suggested ratio found in the ellipsoidal analysis. In that case, the movement presumably causing the orthotropic value was accommodated by localised deformation. This highlights the fact that both true geometry and material properties are necessary. Hence simplified ellipsoidaI shapes may mask the significant part of the left ventricular movement when they are used to model the ventricle for both material property determination and also for stress and deformation analysis.
Medical & Biological Engineering & Computing
January 1979
29
In the above
The solution for 17o is given as
pa 2
az
~/zF I1
Owing to the restricted number of unknowns to form suitable equations, it is necessary to specify one variable. Taking this to be t~0 and using the calculated value of Eo, then the remaining two unknowns can be found as
(a z sin 2 $ + b 2 cos ~ 4,)2 sin r
oJ (a 2 sin z ~ + b 2 cos z ~b)sin
a~
(1 - ~,~o) /
(b 1 1,, t[;{WT-~42h
E. =
lz
J
d4,
E--~ ,
13 =
sin ~b
(4)
(5)
(6)
o
-- {~z --2be F~,olx--2Iz(1--Fc, o)} Eo]
(2)
and Foo ~ ~
30
IJ'r
.
.
.
.
.
.
.
.
.
.
(3)
and P is the increment of applied pressure, a is the major axis half-length (dimension), b is the minor axis halflength (long axis), h is the wall thickness, W~=o is the incremental change in b, W, =90 is the incremental change in a, Eo, E~, are the two moduli of elasticity in the circumferential and longitudinal directions and t~,0, t~0, are the two Poisson's ratios.
Medical & Biological Engineering & Computing
January 1979
