Factors influencing left-ventricular stiffness

Factors influencing A.L. Yettram”,

B.S. Grewal*,

left-ventricular J.R. Dawson**

and D.G.

stiffness Gibson’

Department of Mechanical Engineering, *Brunei University, Uxbridge, Middlesex, UB8 Brompton 3PH, * *Cardiac Department, The London Hospital and ‘Cardiac Department, Hospital, London Received

April

1991, accepted

August

11J91

ABSTRACT 77teaim of the study was to investigate the relative contributions of geometrical and material factors to overall lefiventricular cavity st@iress. Left-ventricular cavity shapes were reconstructed using a computer and the variation of myocardial elastic modulus was calculated, by the finite element method, through the passive phase of diastole when rising volume coincided with rising pressure. Geometric data were obtained from biplane cineangiography, with micromanometer pressure measurements, for ten patients with left ventricular disease. Dimensional analysis was applied to the initial and derived data from which the injluences of myocardial compliance, wall thickness-to-long dimension ratio, and aspect ratio (long-to-short axes) were determined. The ratio between the volume elasticity and the myocardial modulus of elasticity, the normalized str$itess ratio (NSR), is proposed as a use&l index of left ventricular mechanical behaviour in diastole. The volume elasticity of the chamber is dependent not only upon the myocardium elastic modulus and the wall thickness ratio, but also on the shape of the chamber. Changes in the thickness/radius ratio of the ventricle have less effect upon its distention than those in the long dimension/radius ratio. The left ventricle becomes more spherical in shape through diastole and hence becomes stiffer by this geometric mechanism. Keywords:

Left ventricle,

stiffness, myocardium,

volume

elasticity

INTRODUCTION

Table

Measures of cavity compliance or myocardial stiffness are frequently invoked to describe left ventricular diastolic behaviour. However, myocardial stiffness is not the only factor determining the compliance of the chamber. Increased chamber stiffness, for example, may result from wall thickening without any change in the stiffness of the heart muscle itself. It is also possible that a normal wall thickness may be associated with a change in chamber stiffness if there is a change in the ventricular geometry. The purpose of the present paper is to determine some of the major factors which influence the overall stiffness of the left ventricular chamber, independent of the properties of the myocardium itself.

Patient

METHOD Ten patients were studied, two of these were studied on two separate occasions. The diagnoses are given in Table I. In all cases, cardiac catheterization was required on standard clinical indications. ams were perfomed during Biplane cineangio held mid-inspiration !r60 degrees left and 30 degrees right anterior oblique projections) at a frame rate of 5Os-‘, with 50ml of Urografin 370 at a flow rate of 12 ml s-’ injected into the ventricle. The left ventriculograms were calibrated by displacing the catheter table by 10 cm. Correspondence

and reprint

0 1992 Butterworth-Heinemann 0141-542~5/!~2/010021-0ti

requests

to: Professor

A.L. Yettram

MO AN RE BA HA MI CU WE WE(A) &A) FO

1

Details of patients

studied

Diagnosis

Sex

Age

Congestive cardiomyopathy Mild mitral stenosis with regurgitation Coronary artery disease Coronary artery disease with left ventricular dysfunction Coronary artery disease Coronary artery disease Coronary artery disease Coronary artery disease Coronary artery disease with induced angina Coronary artery disease Corona+ arte< disease with induced angina Aortic valve dysfunction with LV hypertrophy

F M M

60 27 60

M M M M M M M M F

46 55 45 70 47 47 57 57 68

Ventricular pressure was measured using a Gaeltec tip micromanometer, mounted on the angiographic catheter, using atmospheric pressure as zero. It was immersed in saline for 20 min before use, to minimize zero drift, which was found to have been less than 2 mmHg by the end of the study for all patients. The transducer was calibrated in situ against eak systolic and minimum diastolic pressure recorde B through the angiogra hit lumen. Pressures were recorded photographica Ply at a paper speed of 0.1 m s-’ with a simultaneous electrocardiogram. DATA ANALYSIS Beats analysed, occurred within the first five after the

for BES J. Biomed.

Eng. 19’32, Vol. 14, January

21

Fac6ors injlwncing left-ventricular st~fkss: A.L. Yettram et al.

start of contrast injection, when the haemodynamic effects of the agent itself are negligible, and were not extrasystolic or postextrasystolic. The onset of the QRS complex of the electrocardiogram was used to gate a light emitting diode and thus identify its timing on the tine film. Therefore the pressure was synchronized with the tine. The cineangiograms were digitized in both planes, frame by frame’ throughout the cycle being studied. The timing of mitral opening was taken as the first appearance of unopacified blood through the mitral valve at the start of diastole, best identified on the right anterior oblique projection. Cavity volume was derived for each pair of right anterior oblique (RAO) and left anterior oblique of the (LAOk frames by computer reconstruction cavity . The two frames were aligned at the apical point, taken as the point furthest from the mid-point of the aortic root on each. The RAO view was then sliced perpendicular to the mid-aortic to apical line at 50 levels, thus providing 100 points defining its outline. These points were then projected across to the LAO view and the intersections with its periphery were recorded. Using computer-aided surface fitting techniques the three-dimensional cavity shape was reconstructed and its enclosed volume obtained. This method has previously been validated directly against plaster casts of left” and right” ventricles. The pressure/volume curve for each patient was established, and that part of diastole considered as passive (rising volume with rising pressure) identified, until onset of the ‘a’ wave, as illustrated by Yettram et aL4. The slope of the P/V curve at any instant is a measure of the ventricular, i.e. cavity, stiffness. Using these curves as data the variation of myocardial elastic modulus over the relevant range was calculated for each patient. This was done as follows. It was assumed (i) that the myocardium was linearly elastic over each inter-frame interval of 0.02 s; (ii) that small deflection theory would obtain over such a small time step; and (iii) that the myocardium was orthotropic with a modular ratio of along-fibre/cross-fibre of 2 to 1 (Yettram and Vinson”). In the finite element model the along-fibre direction is considered to change from 60” to the circumference at the epicardium to -60” at the endocardium, following the work of Greenbaum et al.“, and it is assumed that the modular ratio stays constant over the limited range of the cycle being investigated. Details of the finite element idealization are given by Yettram et aL7. Using arbitrary elastic properties an initial frame model was set up and analysed for the increment of pressure between the first and the second frames. The deformed shape was obtained by adding the displacement to the original geometry and from this the new cavity volume was calculated. The displacements were then scaled to produce a match of the new cavity volume with the actual cavity volume for the second frame and thus, the actual major and minor moduli for the myocardium, between the first and second frames could be obtained. The process was then repeated for successive pairs of frames. The incremental approach that has been used here may not be as accurate as the use of a geometric stiffness matrix based on higher order expressions for strain in terms of displacements, but in the context of this investigation and the assumptions of linear elastic and

22

J. Biomed. Eng. 1992, Vol. 14, January

small deflection behaviour between closely frames, it is adequate for the purpose.

spaced

RESULTS The results of the present analyses are shown in Figure 7 for the 12 cases where the major ‘alon fibre’ modulus is plotted against frame number. d e nonconstant nature of the myocardial stiffness is very clear in all cases. It should be noted, of course, that all of these patients differed in their heart rate and also in the region of diastole where the myocardium showed passive properties. Table 2 provides data with regard to the ranges over which Figure 7 applies. A linear relationship is assumed to exist between frames so as to extract approximate values for filling rate (d V/dt), rate of ressure rise (dP/dt) and ventricular compliance PdV/dP). ANALYSIS Consideration of the method used and the assumptions upon which it is based, implies a linear relationship between chamber stiffness and myocardium modulus of elastici i.e. stiffness, for each patient. Hence an initial an at?isis was made by applying a least squares regressional fit to data obtained for each patient at mid-frame, and the relationship between myocardial and ventricular stiffness, represented by an equation of a straight line such that

can be obtained where E represents the minor modulus of elasticity. A large variation in slope, m, shown in Table 3, is clearly evident. Hence it would seem that dP/d V cannot be used for the comparison of myocardial stiffnesses between ventricles. This argument is discussed by Mirsky and Parmley” and Gaasch et al. “I. They propose that differences in ventricular size should be accounted for by defining the term ‘volume elasticity’ as V(dP/d V). Therefore we investigated the relationship between the passive elastic myocardial stiffness E and the volume elasticity V(dP/dV). It has been suggested, Mirsky ” and Gaasch et al. lo, that a relationship exists between these parameters and also involves the wall mass. If V(dP/dV) is plotted against E for each individual case studied here and best straight lines fitted to each, the results can be expressed as values for the slope and the intercept for each patient, and are given in Table 4. The results are given in order of slope magnitude so that a direct comparison can be made with the results for the passive modulus versus ventricular stiffness values E/(dP/d V) in Table 3. There appears little evidence from results here that volume elasticity versus myocardial elastic modulus accounts for a complete qualitative explanation of why an individual chamber distends in the manner that it does. So the question that arises is what other factors might play a significant role in this? Gibson and Brown” argue that the ventricle’s capacity to fill may depend not only on the distensibility of the myocardium but also on the configuration and change in cavity shape.

Factors in&ruing

6 ‘S 5

0'

I

23

I

24

I

25

26

’ 27

01

37 38 39 40 41 42 43 44

29

30

31

32

33

34

58

59

60

61

A.L. Yettram

’

et al.

’

29 30 3

cu

AN

57

stifiss:

0lfi.l.I

26 27 28 29 30 31 32

RE

MO

‘m‘

o- 25

I$-uentricuhr

46

48

50

52

54

56

30 31 32 33 34 35 36 37 38

s

MI

FO

._ m

32 33 34 35 36 37 38 39 Frame

Frame

WE(A) Figure 1 Variation

27 28 29 30 31 32 33 34

number

elastic modulus

Table 2 Data regarding the range in diastole corresponding to start and final frames chosen Patient

34

Heart rate

studied

Total number of frames

35 Frame

number

36

37

38

29 30 31 32 33 34 35 36 Frame number

number

HA

BA

CL(A) of major myocardial

CL

WE

against

frame number

for each

Onset

case.

of passive

for the 12 cases studied

X,, X, = percentages

filling

of cardiac

End of passive

cycle

from

Frame number

Volume (cm”)

Frame number

Volume (cm.0

22

!)7 8X

:1I :14

28

l’L2.0 105.X

27 34

170.7 173.4

z

94 77 41) 63 xx 7.5 50 47

x 39 61 4x 34 40 60 64 47 38

24 2x 5ti 29 26 2x 45 81 36 33

63.ti 71.1 I I lI.(i 161.5 161.2 89.5 124.8 8X.3 63.2 146.0

:I2 36 61 38 34 3x 56 3!) 44 38

121.7 Xfj.fi 157.5 195.5 234.8 1’29.5 l’Lfi.9 126.9 84.4 177.7

Dimensional

6.4 7!)

analysis

of the left ventricular

chamber

The need to know why patients exhibit unique individual patterns of a straight line relationship between not only chamber stiffness and myocardial modulus but also volume elasticity and myocardial modulus re uires further explanation. It was decided to isolate va‘1ues of dP/d V, E, t, r, and 1 (where t, r, and 1 are the wall thickness, radius and long dimen-

of QRS

complex

filling

MO AN

E. CL(A) HA WE WE(A) FO RA

onset

sion typically) for each and all of the patients and then to apply an appropriate means of analysis which would attempt to find a relationship between dP/d V and the other parameters. We therefore undertook a more complex procedure which would include wall thickness and both long and minor axis dimensions as parameters. Thus a dimensional analysis study based upon these factors, thought to affect left ventricular filling in

J. Biomed.

Eng. 1992, Vol. 14, January

23

Factors injhncing

.!@ventriculur stiflsless: A.L. Yettram et al.

Table 3 Constants for the linear relationship stiffness (dP/dV) and the myocardial minor (Equation 1) m (kNmmmHg_‘)X

Patient

between modulus

WE(A) WE AN RE HA CL MO &A) BA

Table 4 Constants for the linear relationship elasticity (VdP/d V) and the myocardial minor Patient

MI BA HA MO E(A) AN CL WE %(A) FO

-2.53 -0.46 -0.79 - 0.36 -0.86 -2.99 - 7.39 -0.63 - 15.86 0.74 - 17.03 - 14.40

between the volume modulus of elasticity

Slope (mmHg/kNm-‘L)

Intercept

0.57 0.59 0.86 0.96 0.97 0.98 1.15 1.27 1.31 1.32 1.54 2.01

0.21 5.49 3.38 10.09 1.07 8.64 0.81 0.45 0.01 1.67 0..53 2.36

(mmHg)

diastole was carried out (Grewal’“). Using the Btickingham Pi Method, and a statistical package, namely GLIM (generalized linear interactive model), this results in a final expression of the form dV -=dP

r3

t

(2)

E 4-I 1

Thus such a form of analysis leads one to speculate that not on1 is the compliance of the chamber (d V/ dP) depen drent upon the passive elastic myocardial stiffness (E) but also on its geometric parameters, i.e. the inner chamber radius, and the ratio of the thickness of wall to the long dimension of the chamber. The ratio of E to dP/ud V, the value of m in Table 3, is in units of length cubed (L”). A form of nonlinear regression was attempted upon the values for dP/d V, E, t, r and I, with the input or independent variables being the minor modulus, radius, long dimension and thickness of myocardium. The applicable model was of the form, as proposed by Draper and Smith’” Y= &@x2yx;+E

(3)

or in our application

g=f(E,t.rJ

d,YA+&

Subsequent analysis, taking all patients resulted in the following relationship:

24

j. Biomed.

Eng. 1992, Vol. 14, January

This equation, where dP/d I/’ was assumed to be proportional to E, appears to highlight some major findings with respect to the chamber. Since rW5.0071 .‘I t0.7Y2can be broken down into

fkN mm”)

10h

41.96 54.52 82.92 92.37 116.68 123.95 129.60 147.93 165.62 227.66 242.20 300.66

FO cu

the chamber of elasticity

(4 together,

ventricular compliance may also be said to be dependent upon the ratios of length-to-radius and wall thickness-to-radius. This is analogous to stating that dP/d Vis dependent upon a form of shape index (I/r) or aspect ratio and also upon relative wall thickness (t/r). It then follows that any change in ventricular compliance appears more dependent upon change in the long dimensional length than on change in wall thickness. For example, a 10% increase in the long dimension with all other parameters unchanged, produces a 12.20/oincrease in dP/ d V whereas a 10% increase in wall thickness only produces a 7.8% change in dP/d V. The cubic radius term approximates closely to the volume within the chamber, Feigenbaum et aZ.‘5. For our patients, the correlation coefficient between these was found to be 0.98. Therefore equation (5) is used in determining the degree of ‘goodness of fit’ via the regression coefficient through back substitution of the actual midframe chamber stiffness values. This results in a correlation between the observed and obtained values having a regression coefficient equal to 0.996. Inter-relationship between myocardial elasticities

chamber

and

Observation of the change in the ratio of the long dimension to the inner chamber radius of the left ventricle (Z/r) is one simple method that has been used to study how the chamber changes sha e throughout the cardiac cycle (Phillips et al. P6, Gaasch17). Equation (5) can be rearranged such that (“:“,(

t”:i”, (6)

when the power to which r is raised is rounded to -5. Since the left ventricular volume is directly proportional to the cubic term for the inner chamber radius, then for each patient

where k is a new constant. This latter expression, whilst including the volume elasticity term, suggests that geometric parameters, namely long axis and wall thickness, play a major role in determining ventricular compliance. In particular the volume elasticity depends not only upon the myocardium stiffness but also on the geometric configuration of the chamber. Passive deformation of the chamber is thus a function both of the physical and geometric properties, which can influence each other such that no single entity may be regarded as independent. The term on the left

Factors injluacing

of equation (5), namely “,dp

E

dV

can be termed the normalized stiffness ratio, or NSR, since it denotes the volume elasticity per unit of myocardium stiffness (or in other words as the normalized ventricular chamber stiffness divided by the myocardium modulus). DISCUSSION It is obvious that volume changes during filling must be associated with corresponding alterations in dimensional and geometric parameters such as wall thickness, radius, long dimension and regional wall movement. Most expressions said to be indicative of stress levels or elastic stiffness include such parameters. Hence, any analysis which attempts to describe quantitatively left-ventricular expansion during diastole, and for different cardiac states, would appear to require the need for a more detailed examination than simple volume estimates and instead be based upon a form of dimensional analysis. The analysis should include at least the geometric and dimensional parameters we have discussed. Since the size and shape of the ventricular chamber is a determinant of the stress that is present in the myocardial fibres at any intraventricular pressure, it therefore appears that geometry must be of primary importance in cardiac function. Hence, some doubt must be cast upon the reliance of any findings which are merely based upon sole consideration of pressure-volume characteristics of the chamber and the claim that that curve alone is indicative of the myocardial properties. The results presented illustrate the intimate interrelationship between the volume elasticity, the myocardial stiffness and the geometric dimensions of the left ventricle in accounting for its behaviour. Some general conclusions can be drawn from these results. Clearly the volume elasticity of the chamber is dependent not only upon the myocardium modulus but also upon the shape and nature of the chamber. It is also significant that changes in the thickness/radius ratio of the ventricle at any instant in diastole have less effect upon the distention of the chamber than those in the long dimension/radius ratio. Various cardiac diseases can potentially affect the mechanical properties of the myocardium directly and also the pattern of deformation and hence the interrelation between the two. In the patients studied the effect of pericardial restraint was investigated but there were no indications of any external constraints being present so as to affect chamber shape change. Also we found no relation between stiffness and filling rate, indicating that viscous effects were not important in the phase of diastole which we studied. By contrast, our approach emphasizes the potential clinical im ortance of the interrelationships between the variab Pes which are involved. A relation may be completely determinate as, for example, that imposed by a constant myocardial mass. Alternatively, it ma be less well defined and more at the biological leve r , in that hypertrophied ventricles are usually elongated

Icft-ventricuh

st@zess: A.L. Yettram et al.

whilst wall thickness of those with a spherical configuration are almost invariably normal. Similarly, impairment of normal functioning in the isovolumic relaxation phase, particularly by this leading to a change in cavity shape, could affect the subsequent filling of the chamber and hence alter the geometric configuration in late diastole. The chamber stiffness and the volume elasticity are both directly proportional to the myocardial stiffness and the effect of disease upon the left ventricle is such as not only to affect the volume elasticity or the myocardium elasticity individually, but the manner in which they interact with one another; in effect the constant of proportionality between the two. These ideas can be taken into account by considering, as an index of left ventricular function, this constant of proportionality which is the dimensionless parameter

V

dP

E

dV

-x--

(the normalized stiffness ratio), rather than any derived from an assumed exponential relationship existing between pressure and volume. Being dimensionless throughout diastole, the NSR therefore has properties which allow values to be compared between patients with different ventricular sizes, and thus provides a framework for assessing overall diastolic function. CONCLUSIONS The model used is simplified in many respects, e.g. in assuming that the myocardium of the ventricle is homogeneous (although not isotropic). However the procedure used with a model which is geometrically representative of real left ventricular shape, as opposed to highly idealized forms such as ellipsoids which are used by other investigators in the field, and based on real human data is a step forward in endeavouring to determine how myocardial stiffness changes through mid- to late diastole. The procedure produces an overall ‘average’ measure of m ocardial elasticity as we are dealing with a one- cyegree-offreedom system essentially, i.e. matching overall elasticity against overall enclosed volume change. It is our hope to produce a more detailed description of the myocardium, as a multi-degree-of-freedom system which would involve matching against parameters other than just volume, e.g. long dimension, diameter, wall volume, modular ratio and enclosed volume change all simultaneously. REFERENCES Gibson DG, Prewitt T A and Brown DJ. Analysis of left ventricular wall movement during isovolumic relaxation and its relation to coronary artery disease. Br HeartJ 1976; 38: 1010-9. Yettram AL, Vinson CA and Gibson DG. Computer modelling of the human left ventricle. Tr ASME 1982; 104: 148-52. Redington AN, Gray HH, Hodson ME, Rigby ML and Oldershaw PJ. Characterisation of the normal right ventricular pressure-volume relation by biplane angiography and simultaneous micromanometer pressure measurements. Br Heart J 1988; 59: 23-30.

J. Biomed. Eng. 1992, Vol. 14, January

25

Faders infwncing I@-ventricularstajbxs: A.L. Yettram et al. 4. Yettram AL, Grewal BS, Gibson DG and Dawson JR. Relation between intraventricular pressure and volume in diastole. Br Heart J 1990; 64: 304-8. 5. Yettram AL and Vinson CA. Orthotropic elastic moduli for left ventricular mechanical behaviour. Med and Biol

Eng and Comp 1979; 17: 25-30. 6. Greenbaum RA, Ho SY, Gibson DG, Becker A and

Anderson RH. Left ventricular fibre architecture in man. Br Heart J 1981; 45: 248-63. 7. Yettram AL, Vinson CA and Gibson DG. Effect of myocardial fibre architecture on the behaviour of the human left ventricle in diastole. J Biomed Eng 1983; 5:

321-8. 8. Oden JI. Finite Elements of Nonlinear Continua. New York: McGraw-Hill, 1972. 9. Mirsky I and Parmley WM. Evaluation of passive elasticstiffness for the left ventricle and isolated heart muscle. In: Mirsky I, Ghista DN and Sandler H eds, CardiacMechunics, New York; Wiley, 1974, 331-58. 10. Gaasch WH, Apstein CS and Levine HJ. Diastolic, properties of the left ventricle. In: HJ and Gaasch WH eds, 7;lre Ventrick, Boston: Martinus Nijhoff, 1985,

26 J. Biomed. Eng. 1992, Vol. 14, January

143-70. 11. Mirsky I. Assessment of diastolic function: suggested methods and future considerations. Circulation 1984; 69 836-41. 12. Gibson DG and Brown DJ. Continuous assessment of left ventricular shape in man. Br Heart J 1975; 37: 904-10. 13. Grewal BS. The mechanical behaviour of the left ventricle of the human heart in diastole. [PhD Thesis]. UK: Uxbridge, Brunel University, 1988. 14. Draper NR and Smith H. Applied RegressionAnalysis, New York: Wiley, 1981, 458 et seq. 15. Feigenbaum H, Popp RL, Wolfe SE, Troy BL, Pombo JF, Haine CE and Dodge HT. Ultrasound measurements of the left ventricle: a correlative study with angiography. Ann Znt Med 1972; 129: 641-50. 16. Phillips CA, Grood ES, Schuster B and Petrovsky JS. Left ventricular work and power: circumferential, radial and longitudinal components. Mathematical derivation and characteristic variation with left ventricular dysfunction. JBiomech 1982; 15: 427-40. 17. Gaasch WH. Left ventricular wall ratio. Am J Cardiol 1979; 43: 1189-93.