BUT.T.ETIN OF

~THEMATXCALBIOLOCY VOLU~E 38, 19"/6

PASSIVE ELASTIC W A L L STIFFNESS OF T H E L E F T VENTRICLE: A COMPARISON B E T W E E N L I N E A R T H E O R Y AND LARGE DEFORMATION T H E O R Y

I. MmSKY, I~. F. J~z,% B. R. KUB~.RTt, B. KORECKY~AND G. C. T ~ c m ~ x ~ Department of Medicine, Harvard Medical School and Peter Bent Brigham Hospital, Boston, Massachusetts. r Systems Analysis Section, The Aerospace Corporation, Los Angeles, California, U.S.A. :~Department of Physiology, The Faculty of Medicine of the University of Ottawa, Canada

Assuming a spherical geometry for the left ventricle, passive elastic stiffness-stress relations have been obtained on the basis of linear elasticity theory and large deformation theory. Employing pressure-volume data taken from rat hearts of various age groups, it is shown that young rat heart muscle (1 month) is stiffer than either adult (7 months) or old rat heart muscle (17 months). Although the qualitative results are similar for both elasticity theories, the large deformation theory gave results in closer agreement with those obtained from papillary muscle studies. These results imply that stiffness of muscle per se can be assessed from left ventricular pressure-volume data.

Introduction. In recent years there has been an increased interest in the mechanical behaviour of the left venticle during diastole and in particular, the assessment of ventricular stiffness utilizing the pressure-volume relations (Dodge et al., 1962; Bristow et al., 1970; Hood et al., 1970; Diamond et al., 1971, 1972; Forrester et al., 1971, 1972; Gaasch et al., 1972; McCullagh et al., 1972; Barry ctal., 1974). Most of these studies relate to simple ratios between pressure and volume. While simple ratios such as AP/AV, d P / d V i.e. instantaneous change in pressure with change in volume, end diastolic pressure divided by 239

240

I. MII~SKY, 1~. F. JANZ, B. R. K U B E R T , B. K O R E C K Y AND G. C. TAICHMAN

end diastolic volume (EDP/ED V) etc. m a y be useful in detecting changes in ventricular stiffness, they do not quantitate wall stiffness. Since wall stiffness is a continuously varying function of wall stress (Mirsky et al., 1973a, 1974), terms such as d P / d V or AP/A V are meaningless unless they are evaluated at a given pressure. When elastic stiffness is defined on the basis of the stress-strain concept, comparisons between ventricles of different size and shape are made more meaningful. The purpose of this study is to compare the qualitative and quantitative results of elastic wall stiffness based on the linear theory of elasticity (Mirsky, 1973a, 1974) with those based on a large deformation theory. The essential difference between the present large deformation theory and those developed previously b y Mirsky (1973b), Janz et al. (1974) lies in its simplicity for application in the clinical situation and hence its attractiveness to cardiologists.

Definitions. Some of the terms employed in this study will be first defined since inconsistent terminology has often led to confusion in the literature on cardiac mechanics. Stress--denoted here b y a m a y be defined as force per unit cross sectional area of a material and essentially is a measure of the intensity of forces.

Strain--denoted here b y e represents a change in length with respect to a reference length. In the present discussion we define an increment of strain to be de = dl/l i.e. an instantaneous change in length with respect to the instantaneous length l. This is consistent with the natural strain definition given b y e -- log 1/lo where l0 is the zero stress length.

Elastic stiffness E is defined mathematically b y the quantity da/de i.e. an instantaneous change in stress with respect to an instantaneous change in strain. The units of stiffness are the same as those of stress i.e. g/cm 2 or dynes/ cm 2.

Stretch ratio ,~ represents the ratio of the instantaneous length and zero stress length i.e. ~ -- l[lo. Theoretical Considerations.

In this analysis, we are primarily concerned with the elastic stiffness of the ventricular muscle per se and therefore the ventricular geometry is assumed to be spherical. Although ventricular geometry affects the quantitative results for wall stress and hence wall stiffness, Mirsky etal. (1974) have shown that the elastic stiffness constant k (E = /ca + c) is unaffected by the geometry.

Expressions for elastic stiffness based on linear theory. Based on previous analyses (Mirsky et al., 1973a; Mirsky, 1973b), the elastic stiffness Em at midwall

PASSIVE ELASTIC WALL STIFFNESS OF THE LEFT VENTRICLE

241

radius R and "incremental modulus" E I ~ c may be expressed in the following forms: Em = (6 VR/a)(b2/a 2 + b2)[(am + Po)(dP/dV)/PT + (am + P ) / ( V + Vw)] + 3a2(b3 - a3)(am + P)/(a 2 + b2)(as + 2R 3)

(1) E I ~ c = (3a/2)[(a 3 - ba)/b(a 2 + b2)

{1 + (V/Pr)(dP/dV)}{1 + (Vw[V)a2/(a 2 + b2)}]

(2)

where V is the instantaneous left ventrieular volume, Vw is the wall volume, P and P0 are the pressures acting on the internal and external radii a, b respectively of a sphere and PT = P - P0 is the transmural pressure. The midwall circumferential stress am and a (the difference between the midwall circumferential and radial stress components i.e. a = a c - at), may be written as (Timoshenko and Goodier, 1951) am -~ PT(1 +aS/2Ra)(1 + V / V w ) - P a = (3PT/2)(V/Vw)(bS/R 3)

(3)

Equations (1) and (2) are to be employed later for comparison purposes. Pressure-volume relations based on large deformation theory. For a sphere, the stress equation of equilibrium may be written as (Green and Adkins, 1960) damD/dp + (2/p)(amm - a0o) = 0

(4)

where atom, nee are the radial and circumferential stress components referred to a deformed radius p. Consider now a modified form of the stress-stretch relations first suggested by Blatz et al. (1969), namely, atom = H(p) a o o = F(~o)+ H(p)

(5)

where F(~e) is as yet an undetermined function of the stretch ratio 4o = pit (the ratio of deformed to undeformed radius) and H(p) is a "hydrostatic pressure." This representation differs from a similar one employed by Janz et al. (1974) in several respects, (a) ventricular muscle behaves more as an anisotropic elastic material rather than an isotropic material assumed in the studies by Janz et al. (1974), (b) they uncouple the radial direction from the circumferential direction which is consistent with the fibrous structure of the left ventricle, and (e) as will be observed later, the numerical analysis is greatly simplified and readily applicable to the clinical situation.

242

I. MIRSKY, R. F. JAN'Z, B. R. K U B E R T , B. K O R E C K Y AND G. C. TAICHMAN

Direct integration of (4) with the aid of (5) yields the following relation for the transmura! pressure, namely, A P = aDD(b)-- apD(a) = - P o + P = 2 f : F ( g o ) dp/p

(6)

If use is made of the incompressibility condition 2~22e = 1 where 2p = dp/dr and 20 = p[r (Green and Adkins, 1960), the above expression is transformed to AP =

2 ~;tolb)

F(2o) d2o[2o(1 - 2~)

(7)

,) ~o(a)

This is the integral form for the pressure-volume relation since go(a) = a[ao = (V/Vo)I/a and

(V+ Vwy/'

2o(b) = b/be = \-V--o-o+'V--w] where the zero subscript quantities correspond to the state of zero transmural pressure. It is convenient here to assume an infinite series expansion for the function F(go) in the form F(20)-- 28 ~ an(2~-l)n

(8)

n~0

where the constants an are determined from known pressure-volume data. If in particular, the pressure-volume relation is expressed in the exponential form AP = A + B exp C ( V - Vo) (9) it is shown in Appendix A that the function F(2o) may be approximated by the expansion F(2o) ..~ 2gV3(A +B)/2 loge (1 + Vw/Vo) k + (3BCVo[2)(2~-1)exp {(20s - 1)CV0} +(3B/2)

~

(CVo)nczn(2~ - 1)n/(1-an)(n - 1)!

(10)

for sufficiently large N. (a = Vo/[Vo+ Vw]). In the actual computations only three terms were necessary. Uniaxial stiffness-stress relation. Consider the three-dimensional stressstretch relations implied by (5) to be of the form

PASSIVE ELASTIC WALL STIFFNESS OF THE LEFT VENTRICLE

243

al = Hip)

a, = H ( p ) + F , ( , ~ , )

i = 2, 3

(11)

I f we assume that the material is under a state of uniaxial stress, then al = 0, a2 = 0 and the above relations reduce to a1 = H ( p ) = 0 a2 = F2(22) = H ( p ) = 0

i.e. Fg.(A2) = 0

as = a = F s ( A s ) + H ( p ) = F ( I )

(12)

where F(A) is defined by (10). Defining elastic stiffness on the basis of the natural strain definition, we obtain E = da/d8

(with ~ = log e A)

= ,~ d a / d A = I dF/dJl

(13)

It is this expression which is plotted against F in order to yield the uniaxial stiffness-stress relation. Methods Experimentalprocedure. An automatic servocontrolled system was developed to determine small changes of pressure and volume in left ventricles of rats from three different age groups (Koreeky et al., 1974). The heart of a closed chest anesthetized rat was arrested by injecting KC1 into the jugular vein and subsequently perfused with low Ca Krebs-Ringer solution at approximately 10~ After thoraeotomy, the heart was cannulated through the aortic valve by a vinyl catheter to which the root of the aorta was secured. After the right ventricle was slit open, the heart with the cannula was removed from the thorax and mounted into a pressure-volume set-up filled with saline. The continuous P - V changes were obtained from this closed system b y amplifying the output signals of pressure and volume transducers and displaying them on an X - Y recorder. The direction of the constant flow was reversed b y pressure sensors preset to the required maximum positive and negative pressures, as well as b y a back up volume sensor. Pressure sensors were checked by a calibrator and a reference point for zero transmural pressure was provided by a bath level indicator. The volume at zero tra~smural pressure was determined directly from the pressure-volume diagram. Pressure-volume relationships were obtained with the volume changes going from low to high and then back down three times and each experiment was completed within 15-20 min so that development of rigor mortis was minimized.

244

I. MIRSKY, R. F. JANZ, B. R. K U B E R T , B. K O R E C K Y AND G. C, TAICHMAN

Pressure volume r e l o t i o n s in rot heorts

I0.0

! !

--

9.0

!

::~

8.0

E E

7.0

I

I

/

Q_" 6 . 0 5.oo)

!

4.0-

I

Q.

3.0--

E

2.0

I--

I,C

/ /

I 0

Figure

1.

i

j

ol

0.1

I~o

0.2 0,3

Pressure-volume

o'-"o YOUng( I mo.) o-"OAdul f (7mo5.) I~'-'-e 0 [d (17mos.)

I

[

I

I

0.4 0.5 0.6 VoI. V~ mL

0.7

I 0.8

1 0,9

r e l a t i o n s f o r r a t h e a r t s o f t h r e e d i f f e r e n t age

groups. T h e d o t t e d a n d c o n t i n u o u s lines r e p r e s e n t t h e e x p o n e n t i a l c u r v e fits w h i c h are v a l i d o v e r t h e r a n g e 1.5 m m H g < A P < 10 turn Hg. A t lower pressures, t h e c u r v e s t e n d to b e sigrnoidal i n c h a r a c t e r . T h e s t a n d a r d errors are i n d i c a t e d b y t h e b r a c k e t s (see T a b l e I)

TABLE I Mean Pressure-Volume Data for the Three Age Groups Y o u n g (1 m o n t h )

AP

1" S.E. of mean.

Old (17 m o n t h s )

V ml.

V ml.

V rnl.

rmn Hg 0 0.88 1.76 2.65 3.53 4.41 5.29 6.18 7.09 7.94 8.82 9.71

A d u l t (7 m o n t h s )

0.180 0.204 0.234 0.259 0.278 0.292 0.302 0.311 0.317 0.323 0.328 0.332

• • • • • • • • • • •

0.002t 0.005 0.007 0.009 0.010 0.010 0.011 0.011 0.012 0.012 0.012

0.340 0.405 0.480 0.550 0.607 0.650 0.684 0.709 0.729 0.745 0.758 0.770

• • • • • • • • • • •

0.003 0.006 0.010 0.013 0,015 0.017 0.018 0.019 0.019 0.019 0.020

0.354 0.417 0.495 0.571 0.633 0.681 0.714 0.740 0.760 0.776 0.789 0.800

• • • • • • • • • i •

0.003 0.007 0.009 0.011 0.012 0.013 0.014 0.015 0.015 0.015 0.016

PASSIVE ELASTIC WALL STIFFNESS OF T H E L E F T V E N T R I C L E

245

Numerical Results and Discussion. Figure 1 and Table I represent the mean values of transmural pressure-volume relations obtained from rat hearts of three different age groups. These included 9 young rat hearts (mean age 1 month); 9 adults (7 months) and 10 old rat hearts (17 months). Also displayed in Figure I are the exponential curve fits expressed in the form (9). The relations for these three age groups are given b y AP

=

0.62+0.313 exp (22.3 ( V - V 0 ) )

(Young)

AP

=

0.65+0.355 exp (7.52 ( V - F0))

(Adult)

AP = 0.82+0.282 exp (7.73 ( V - V 0 ) )

(Old)

(]4)

In each ease the epicardial pressure P0 was assumed to have a constant value of 20 cm HuO (1.47 mm Hg). The relations (14) are valid over the pressure range 1.5 mm I~Ig < AP < 10 mm Hg since the P - V curves are sigmoidal at the lower pressure levels. I n c r e m e n t a l modulus-stress r e l a t i o n s 500-(linear theory ) 450 -c,--..o You ng

/

B--.--B AduLt 400 -- ~ Old

/ //

u ~m 350--

/

'//

III

uZ 300

,~//

m -~

/,

f 250 i

//

20C

/

IOC 50

0

/

l

4

1

I

.I

8 12 16 S t r e s s O'lNc ~

I

I

24 g/cm ~ 20

f

28

F i g u r e 2. I n c r e m e n t a l m o d u l u s - s t r e s s relations a t t h e m i d w a l l for t h e young, a d u l t a n d old r a t h e a r t s b a s e d o n t h e l i n e a r t h e o r y of elasticity. T h e relations axe a p p r o x i m a t e l y l i n e a r a n d are g i v e n i n t h e f o r m Ezzve= k a z o o + c w h e r e t h e stiffness c o n s t a n t s k r e p r e s e n t t h e slopes o f these lines. A t a c o m m o n stress level a~Ne, t h e stiffness is h i g h e s t i n t h e y o u n g r a t h e a r t muscle, t h e r e b e i n g l i t t l e difference b e t w e e n t h e a d u l t a n d old h e a r t

246

I. MIRSKY, R. F. JANZ, B. R: K U B E R T , B, K O R E C K Y AND G. C. TAICHMAN

Stiffness-stress relations (1), (2) (linear theory)and (13) (large deformation theory) are graphically displayed in Figures 2-5 on the basis of these P - V relations and the following values for V0, Vw : V0 = 0.180 ml, V w = 0.464 ml (Young); V0 --- 0.340, Vw = 1.154 (Adult); V0 = 0.354, Vw = 1.253 (Old). Midwall stiffnes-sfress relations for r a t hearts ( l i n e a r t h e o r y ) 8OO ~E 720 -o '~" 640 --

o--..o You n g o-..-o Adult z : Old

/ 2"*" /

-

E bJ

560

,j'S~'"

480

_

c 400

,,'2J /

~, 32C (J

24C

./

-

eZ

16C 80

I 0

2

I

I

f

I

I

I

I

J

4

6

S

I0

12

14

16

18

Stress c%,

Figure

3.

Midwall

g/cm 2

stiffness-stress relations

based

on

the

definition

dam/dem (linear t h e o r y ) . The relations are a p p r o x i m a t e l y linear a n d similar t o t h o s e p r e s e n t e d in F i g u r e 2

Table II presents the linear regression equations for these stiffness-stress relations. TABLE II Stiffness-Stress Regression Equations Young

Adult

Em = 38.1 a m + 2 7 . 9 Ez,vc = 18.7 ~ x ~ c - 30.9

Era = 31.2 a m + 2 4 . 0 E I N c = 15.4 a x N c - 26.2 E ---- 22.3 a - 3 6 . 3

E

= 28.50--46.5

Old

Em Ex~c E

-------=

33.2am§ 16.3 axNc-- 29.7 23.9 a - - 47.5

Era, J~INC are respectively the midwall elastic stiffness and incremental modulus based on the linear theory of elasticity. E is the uniaxial elastic stiffness based on the large deformation theory, The elastic stiffness constant k is represented in the form/~ =]r Both stiffness (E) and stress (~) are expressed in g/cm ~.

PASSIVE ELASTIC WALL STIFFNESS OF THE LEFT VENTRICLE

247

In general, at a given stress level, the elastic stiffness is higher, the higher the value of k, i.e. the slope of the stiffness-stress relations. The qualitative results obtained from both elasticity theories indicate that young rat heart muscle is stiffer than either adult or old heart muscle, there being no significant difference between the adult and old heart. The apparent close agreement between the incremental modulus E~nc and the stiffness E based on the large deformation Uniaxial s t i f f n e s - s t r e s s relations for r o t h e a r t s ( f i n i t e Cheory) I 550 ~ 50C-

,,o

~176 o---oAdult

9

/ t

o,,

~ ~.

/ /!

/// /

/2

b ~OC

.~

:~

//..I

1

2oc

u

'.E LtJ

5r 0

I

I

I

,I

4 8 12 16 UniaxialsCress~

, I

W

20 24 g/cm 2

I

28

Figure 4. Uniaxial stiffness-stress relations based on large deformation theory. 2 is the stretch ratio i.e. the ratio of the deformed to the undeformed length. The qualitative results are reasonably similar to those obtained from the linear theory

theory (Figure 5) should be considered with some caution since by definition there is a ~ factor appearing in the expression for Einc i.e. Einc = da/2de. Although the linear theory analysis is restricted to small elastic deformations, the quasi-static analysis demonstrates the nonlinearity of the stress-strain relation as evidenced by the approximately linear stiffness-stress relation. In particular, the stiffness constants k expressed by the relation E = da/d~ = ka + c appear to be useful indices for detecting changes in stiffness in hearts of different size and shape. Furthermore, these constants are independent of the

248

I. MIRSKY, R. F. JANZ, B. R. KUBERT, B. KORECKY AND G. C. TAICHMAN

ventricular geometry and therefore are reflective of the passive elastic properties of the ventricular muscle per se (Mirsky et al., 1974). While it is apparent that the quantitative results obtained from the linear theory are open to question, the qualitative agreement with the large deformation theory is an encouraging sign. On the other hand, the more accurate analysis based on large deformation theory is simple to perform and is in closer agreement with the papillary muscle studies of Grimm et al. (1970) (Figure 5). This point must be weighed with the fact that zero stress dimensions, necessary for such an analysis, are not readily available in the clinical situation.

Stiffness-stress relations for adult rat heart muscle based on various theories 700 F / /

I c~ ~ . 500 "~

"~

/

4oo-

/

7/"

// /S /

2/

," u 300 ~ z 200 --

_7///Papillary ~ muscle

tl,/

,d

/

,,~ /

/

/ ~/ / ~ // - " f

//

t

/ ,,./'Ei.~

./ A./

/~ /

,oo;/Z/ /I 0

I I0

5

I I I 15 20 25 Sl-ress ~ g / c m z

I 30

, I 35

Figure 5. The stiffness-stress relations based on the linear theory (Era, E~nc) for adult rat heart are compared with that based on the large deformation theory (E) and papillary muscle. (Grimm et al., 1970). l~ote the close agreement between the large deformation theory and papillary muscle.

The uniaxial stress-stretch relation for the papillary muscleis given by a

=

2.37 (~z8_~-9)

It must be emphasized that the present analysis yields only global values of the elastic properties of heart muscle and in the clinical situation should be confined to the latter part of diastole. Future models for the quantitation of the diastolic properties of the muscle must include nonhomogeneity, fibre orientation, actual ventricular geometry and the effects of viscosity and inertia. The authors wish to acknowledge Drs. Julia T. Apter and William C. Hunter for their kind and constructive criticisms. This research was supported in part by US PHS grants HL 12711-06 and HL 14651-03 from the National Heart and Lung Institute.

PASSIVE ELASTIC WALL STIFFNESS OF T H E L E F T VENTRICLE

249

APPENDIX A P r e s s u r e - v o l u m e relation based on large deformation theory : E v a l u a t i o n of the function F(20) F r o m (7) of the t e x t we obtain the transrnural pressure A P in the form AP

2

=

( ~o(b)

-F'(2o)d2o/2a(1 -- 2o8)

(A1)

J;,o(a) Assuming F(2s) in the form

F(2o) = 2o3 ~ an()to3 - 1 ) n

(A2)

n=0

the aboveintegral reduces to

ao/().o a - 1) + -- an()~o3 - 1 ) n-z d(2o a - 1)

A P = - (2/3) d)-sa

n=1

~- - - ( 2 / 3 ) [ a 0 loge ().03--1)+

an ()103- 1)n/n] 20'

f

n=l

(A3)

J ).oa

Now

~,, = (V/VoWS; ~o~

_ (v+voll,. \ v---o-o-o-o-o-o-o-o~/

Hence

AP = -(2a0/3) loge a+ i b n ( V - Vo) n

(A4)

n=l

where a = V o / ( V o + Vw)

and

bn ---- (2an/3nVon)(1 - a n)

(A5)

I f the p r e s s u r e - v o l u m e d a t a is curve-fitted to an exponential function of the form A P = A + B exp C ( V - V o )

= (A+B)+B

f [C(V-Vo)]n/n!

(A6)

n=l

one can i m m e d i a t e l y obtain the coefficients an in terms of the known constants A, B, C b y direct comparison of the coefficients in (A4) and (A6) w i t h t h e result a0 ---- - 3(A + B ) / 2 loge a

bn ---- BCn/n!

(A7)

an = (3B/2)(CVo)n/(1 - ~n)(n- 1)!

n = 1, 2 . . . .

The coefficients an m a y be written in the alternative form

(3B/2)(CV~ an

--

[

( n - 1)! +

(3B/2)(CVo) n ( n - 1)!

1

]

l-e n-l+I [~n/(l - ~.) + 1]

250

I. MIRSKY, R. F. JANZ, B. R. KUBERT, B. KORECKY AND G. C. TAICHMAN

Hence the function -~0~0) reduces to -F()~o)----~~ a ~ 2 4 7n~l~an('~aa--1)n] = 20a [ - 3(A + B ) / 2 loge a +

~ (3B/2)(ffVo a)n(,~os - 1)n/(n - 1 ) ! ( 1 - a n) n=l

+ (3B/2)(CVo)(A, a - 1) ~ (CVo)n-I(,~, a - 1)n-1](n - 1)!]

(A9)

jloa [ - 3 ( A + B ) / 2 loge g

+ "~ (3B/2)(6'Vo e)'~(~lo3-1)~/(1 - ~'~)(n- 1)! ~-~1

+ (3B/2)(6'Vo)(2~e8-1)

exp(OVo(2,a-1)}]

for N sufficiently large.

LITERATURE Barry, W. H., J. Z. Brooker, E. L. Alderman and D. C. Harrison. 1974. "Changes in diastolic stiffness and tone of the left ventricle during angina peetoris." Circulation, 49, 255-263. Blatz, P. J., B. M. Chu and H. Wayland. 1969. "On the mechanical behaviour of elastic animal tissue." Trans. Soc. Rheology, 13, 83-102. Bristow, J. D., B. E. Van Zee, and M. P. Judkins. 1970. "Systolic and diastolic abnormalities of the left ventricle in coronary artery disease: Studies in patients with little or no enlargement of ventrieular volume." Circulation, 42, 219--228. Diamond, G., J. S. l~'orrester, J. Hargis, W. W. Parmley, R. Danzig and H. J. C. Swan. 1971. "Diastolic pressure-volume relationship of the canine left ventricle." Circ. Res., 29, 267-275. and . 1972. "Effect of coronary artery disease and acute myocardial infarction on left ventricular compliance in man." Circulation, 45, 11-19. Dodge, H. T., R. D. H a y and H. Sandler. 1962. "Pressure-volume characteristic of the diastolic left ventricle of man with heart disease." Am. Heart J., 64, 503-513. Forrester, J. S., G. Diamond, T. J. MeHugh and H. J. C. Swan. 1971. "Filling pressures in the right and left sides of the heart in acute myocardial infarction." iV. Engl. J. Med., 285, 190-193. Forrester, J. S., G. Diamond, W. W. Parmley and H. J. C. Swan. 1972. " E a r l y increase in left ventricular compliance after myocardial infarction." J. Clin. Invest., 51, 598-603. Gaasch, W. H., W. E. Battle, A. A. Oboler, J. S. Banas and H. J. Levine. 1972. "Left ventricular stress and compliance in man : W i t h special reference to normalized ventricular function curves." Circulation, 45, 746-762. Green, A. E. and J. E. Adkins. 1960. "Large elastic deformations and nonlinear continuum mechanics." Chap. 2. Oxford: University Press. Grimm, A. F., K. V. Katele, R. K u b o t a and W. V. Whitehorn. 1970. "Relation ofsarcomere length and muscle length in resting myocardium." Am. J. Physiol., 218, 1412-I420.

PASSIVE ELASTIC WALL STIFFNESS OF THE LEFT VENTRICLE

251

Hood, W. B., Jr., J. A. Bianco, R. K u m a r and R. B. Whiting. 1970. "Experimental myocardial infarction: IV. Reduction of left ventricular compliance in the healing phase." J. Clin. Invest., 49, 1316--1323. Janz, R. F., B. R. Kubert, I. Mirsky, B. Korecky and G. C. Taichman. 1974. "The effect of age on passive elastic stiffness of rat heart muscle." Biophys. J. (April 1976). Korecky, B., P. Bernath, M. Rosengarten and G. C. Taichman. 1974. "Effect of age on the passive stress-strain relationship of the rat heart." Fedn. Proe., 33~ 321. McCultagh, W. H., J. W. Covell and J. Ross, Jr. 1972. "Left ventricular dilatation and diastolic compliance changes during chronic volume overloading." Circulation, 45, 943-951. Mirsky, I. and W. W. Parmley. 1973a. "Assessment of passive elastic stiffness for isolated heart muscle and the intact heart." Circ. Res., 33, 233-243. Mirsky, I. 1973b. "Ventricular and arterial wall stresses based on large deformation analyses." Biophys. J., 13, 1141-1159. - - , p. iv. Cohn, J. A. Levine, R. Gorlin, M. V. Herman, T. H. Kreulen and E. H. Sonnenblick. 1974. "Assessment of left ventricular stiffness in primary myocardial disease and coronary artery disease." Circulation, 50, 128-136. Timoshenko, S. and J. N. Goodier. 1951. Theory of elasticity. New York: McGraw Hill. RECEIVED 1-8-75 REVISED 5-27-75