AMERICAN
JOURNAL
OF PHYSIOLOGY
Vol. 229, No. 4, October
1975.
Prinkdin
U.S.A.
Mathematical
and mechanical
of stress-strain
relationship
modeling
of pericardium
SIMON W. RABKIN, AND PING HWA HSU Department of Medicine, Sectiolz of Cardiology, and De@rtment and Epidemiology, University of Manitoba, Winnipeg, Canada
W., AND PING HWA Hsu. Mathematical and mestress-strain relationship of pericardium. Am. J. Physiol. : 896-900. 1975.-Several mathematical expressions (models) were compared for use in describing the stressstrain (a - C) relationship of pericardium. The expression c = a[eBc - I] was preferred because of its simpler form, theoretical consistency, and “good fit” of experimental data. A method was developed for estimating the precisions of the estimates of the parameters cy and & This approach can have general usefulness in assessing the significance of a change in stress-strain relationship of various soft tissues following different interventions. A mechanical model was formulated for the pericardium which consisted of springs representing the collagen and elastin fibers connected in parallel. It could be simulated by the above equation and could describe the behavior of the pericardium. RABKIN,
chanical
SIMON
modeling 229(4)
biomechanics;
of
statistical
analysis;
spring
FORCE-DEFORMATION
(stress-strain)
METHOD
The numerical from pericardial
data for this analysis were taken manually force-deformation curves which had formed
Preventive Medicine,
Section
of Biostati&s
the basis of a previous report (2 1). The first point was the one at the onset of deformation, and the last point was the one just prior to specimen failure. In between and including these two points, a total of 25 were transcribed. Six intact pericardial specimens and three specimens subjected to elastic tissue digestion were characterized (see Table 1 in a previous report) (2 1). Stress ((T) is defined as force per unit area, i.e., crosssectional area determined prior to deformation and is expressed as kilograms per square millimeter. Strain (E) is defined as the length changes per unit initial length; to the onset (1 - LJ/lo 3 where 1, is the length corresponding of deformation and 1 is the instantaneous length. Five equations derived for other soft tissues are: u = PE2
(I)
u = aE + pE2
(2)
model
relationship is a fundamental property of living tissue (11, 12, 14, 17, 18). Rabkin et al. (21) recently studied this property in pericardium and observed that it had the same nonlinear stressstrain relationship as other soft tissues. This nonlinear relationship would be best described by a mathematical expression because it would provide a concise description, facilitate data collection, and assist further quantitative research (11). In this report several mathematical expressions proposed for the elastic properties of other soft tissue (11, 12, 16-18) were compared for the pericardium, and one expression was selected and justified. For the purpose of comparison, the precision of the estimates of the parameters in the equation was also derived. The formulation of a model to describe a system is also a useful approach because a model can be used to concisely describe the behavior of the system, to gain insight into its behavior under different stimuli, and to predict its behavior (3). Models have been proposed for many aspects of the cardiovascular system (13, 24), but none for the pericardium. A mechanical model will be proposed here in the hope that it can descri be some of its functions and explain some of its d *isorders. THE
of Socialand
u=
a[e BE -
11 (W
(3)
u=
aeBt + y
(18)
(4)
u
=
[a/@(
1
+
v)]
n
[ep’
-
e-+‘] (5)
Equation 5 is modified from Janz and Grim (16) wherein v is Poisson’s ratio (= 0.49 for incompressible tissues) and a is now a parameter to be estimated. Two other equations were also compared : do/de
= a + flu
(11)
do/de
= Ql(l + pa>,
(6) (7)
Equation 3 can be derived from equation 6 as follows: the extension ratio (X) was defined as the length of the tissue under strain divided by the resting length of the tissue (l/lo). From experimental results, it has been observed that the tangent (da/dX) is correlated with the stress (c) (11). As a first approximation, it can be fitted by a straight line: da/dX = P(a + cu)* An integration gives c + a = (l/c)#” with an integration constant c. If u = C* when X = h”, then the equation becomes g = (g* + a) $(h-‘*) - a. By definiwith tion, g = 0 when X = 1. If C* and X* are substituted these theoretical values, the equation reduces to : a($@ - I) - 1) or fl = a(eSE - 1) with E = X - 1 = ;; =_ lo& Th’ IS is equation 3. Note equation 3 is equivalent to equation 5 when u = 0. Each mathematical expression (e.g., Eq. 1-7) was placed in the form Y = f(X, 0) + e, where Y = stress vector, X =
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MATHEMATICAL
AND
MECHANICAL
MODELING
strain vector, 8 = parameter vector which characterized the mathematical expression, and c is the error vector. The least-square method was used to estimate 0, i.e., choosing a 0 vector such that the sum of squares of deviations, e’e, is a minimum. The normal equations were derived by differentiating the equation for e’e with respect to 8, and then the equations were set equal to zero. The estimators of 8 were then solved. For the linear equations the solution was straightforward, but for the nonlinear equations iterative schemes derived from the normal equations were employed (7). The residual sum of squares for each mathematical expression was calculated for comparison of the goodness of A scheme of linearization, together the precision formuIa for the estimates, for equation 3. Equation 3 may be rewritten in the Xi = Ei,yi = a (esxi - 1) $- Ei. This be linearized by a Taylor’s expansion Q! and fl(a~(*), PC”‘) as follows : Fi(*) = + Ei where
with the derivation of is outlined as follows form: for y i = pi and nonlinear model may around trial values of &(*)ZJ”) + p2’O)Zi2’0)
&(*I pp>
= y i - ab~(e&*)zi = a - &I Pz(*) z p - pb> Zil(O’ = &(O)Xi - 1 Zi2(0)
=
897
PERICARDIUM COmpariSOn of SeVera! mathematical stress-strain rdationship
1.
TABLE
pericardial I
z If
% + ii II
-rl I 5 T II
5 II
b
b
b
b
--
-~
a
B
-1.26 0.83
2.46
1.018
0.45
a B i s c
--.393 7.05
2.73
16.9
0.96 -1.80
aC*)xi,jj(obi
is E
lzs
g42 E
Eb
i-
These precision estimates are calculated after the estimates of it and p are obtained by iteration. These estimates are small and within the linearized range (as will be seen later). A simpler method for estimation of a and P in equation 3 was derived as follows. The equation S (a, P) = 2[JJiQ(Efixi - l)]” is differentiated with respect to cy and P* Setting the results equal to zero gives two normal equations:
cy= c YiP - 1) - 1)” CCeSxi C C
XiPi($ri
#iyiP' -
1)
Since a should be equal in the above two equations, the iteration scheme can be designed to choose P such that these two equations are nearly equal. The value which is equal in both equations is cy. Both these methods should give similar results, but the latter method doesn’t give the precision
Cs - -~
0.00818 AO.0037 11.78 +0.98 0.5208 0.01865 +O. 0062 5.10 dzo.37
2.06
0.45
0.4389
0.94
0.0020 d~O.00081 5.19 zto.33
0.55
1.45
1.30
0.47 -0.49
0.40
1.05
0.22
0.095 -1.63
1.96
5.1
with l)]”
0.0666
4.20
a
B
0.00062 zto. 00022 8.35 ~0.42
1.90
a
F
&(eSxi n-2
for
b
A
- 1)
Since this nonlinear equation is linearized, an approximate precision of the estimator fi and $i can be derived from least-squares theory. They are
models
j 0.84
0.28
0.1392 0.00038 *0.00011 7.96 ho.32 0.00993 0.00212 +O .00065 10.04 hO.50 0.05740
0.00085
-0.0216 0.0712 0.01226
0.00669
0.5042
t4.88
8.40
5.34
0.65
46.13
1.79
0.1005 0.1308
10.95 -0.0778 0.4726
4.55 -0.1271 0.3804
0.560
5.20
0.5860
1.1257
!5.98
7.12
0.370
Z81.18
0.41
0.7326
0.09
3.11
0.368
45.65
3.769
0.00294
0.0135
0.2901
7.861
4.85
5.20
3.20
0.414
18.007
2.06
-0.0392 0.1412
0.2155 0.00403
0.0594
7.847
8.0
6.35
0,029
-0.00185 0.01181
0.0172
7.23
4.148
0.00306
0.0290
0.4482
15.79
6.82
0.559
20.18
0.865
7.85
9.45 -0.03154 0.05129
10.10
0.0734
estimate. One may estimate ct! and @ by the latter and use these estimates in the precision formula from the former method.
method derived
RESULTS
Goodness of jt. In Table 1 the estimated values of the parameters a, ,6, y, and the residual sums of squares corresponding to each mathematical expression are listed for six intact pericardia. Over the stress range O-2.5 kg/mm2, which corresponds to a range of length change from 0 to 128 % of resting length (lo), the two polynomial equations were consistently poorer in fit compared to the three exponential functions. The goodness of fit for the exponential functions is demonstrated in Fig. 1 where specimen A is displayed along with equations 3 and 5. Equation 4 was not included because it predicts negative stress for small strains. All three exponential functions have similar residual sums of squares and give similar estimates of the most important parameter, /3. If a choice must be made among these three expressions, equation 4 will be rejected because it is inconsis-
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898
S. W.
-
ACTUAL
0.9 -
Y
0.6 -
% g os+ UJ 0.4 0.3 -
STRAIN FIG. 1. A typical stress-strain curve for a specimen of canine pericardium is shown in solid line. Two broken lines represent 2 exponential equations 3 and 5 which best fit actual data.
tent with a theoretical consideration (0 = 0 when e = 0), it predicts negative stress when strain is small (G < 0 when E = 0), and y is always negative and larger than P. Equation 5 will be rejected because of its complex form and difficulty in the estimation of the parameter p when Q is not estimated simultaneously. Equation 5 may be more useful in multidimensional problems (16). Equation 3 was selected to best represent pericardial stress-strain relationship. As can be seen, it is simple in form, theoretically consistent, a good “fit” to the experimental by only d ata, and is characterized two par ameters. For these reasons it was the only one subjected to further analysis. The comparison of the “goodness of fit” between the first-order (equation 6) and second-order (equation 7) approximation indicated that the second-order approximation does fit better. However, integration of equation 7 gives a complicated formula which is no better than equation 3 and is also rejected at this time. Effect of selection of 8 and A*. In the derivation of equation 3 (see METHOD), U* = 0 and X* = 1 were chosen because of theoretical considerations. Since G* is the stress at a specific extension rate X”, one might choose any point (G*, X*) on the stress-strain curve. For every pericardial specimen, the choice of a point other than (0, 1) results in equations predicting negative stress. This is inconsistent with the stressstrain definition and therefore another reason to define strain as the length change per unit initial length. Determination of I, . The determination of a specimen’s resting length (lo) is difficult. The extent to which variations in this measurement could affect equation 3 was analyzed. Table 2 gives four estimates of cy, /3, and residual sum of squares obtained by deleting none, the first one, first two, and first three data points, i.e., simulation of the situations that lo, II, 12, and 13 are used in place of I,. For each consecu-
RABKIN
AND
P. H. HSU
thin 1 SE tive change there is an increase in /3 which is and an increase 1n QTwhich is more marked. .e resid ual sum of squares declines which is expected because a unimodal equation is being used to fit data which are essentially bimodal. The bias in each consecutive step is about 1 mm. These results are similar in all other sets of data and indicate that roughly measured I, does not affect the estimation of the most important parameter P by more than 1 SE which here is a small one. The average of P in these six specimens is 8.08, compared with the average Young’s modulus of 6.15 kg/mm2 (21). The similar order of magnitude suggests that an estimate of Young’s modulus can be used as an initial value in the iterative estimation of /3 for convenience. Efect of elastic tz’ssue digestion, The three pericardia subjected to elastic tissue digestion had the following parameters of a and p estimated for equation 3: 0.271 X 10m4A 0.101 X lO-4 (+ 1 SE) and 11.24 j= 0.54; 0,021.l =t 0.0059 and 1.87 & 0,15; 0.2242 =t 0.0594 and 4.47 & 0.66. When they are compared to the same specimen not subjected to elastic tissue digestion (B, D, E, in Table 1, respectively), the last two sets show significant changes in fl (in l-tests, P < 0.01). The larger ct after elastic tissue digestion shows a change not apparent on initial inspection of the force deformation curves, This change is similar to that found by Roach and Burton (22) in arteries subjected to elastic tissue digestion. DISCUSSION The necessity to utilize a mathematical expression (model) for the stress-strain relationship of soft tissues has been clearly stated by Fung (1 l)* He has also pointed out (12) that the ultimate mathematical model is still a distant goal. The elastic properties of the pericardium, not the viscoelastic ones, were herein subjected to mathematical modeling because they a.re more easily studied and conceptualized. It has been generally accepted that a good model should be simple, characterized by few parameters and uncomplicated in form; realistic, consistent with theory; and accurate, fits the experimental data well and precisely estimates the parameters. The equation 0 = a[efi” - l] was chosen because it is relatively simple in form, characterized by only two parameters and consistent with the theoretical consideration that when e = 0, u = 0. The first parameter ~1 characterizes the level of strain at which stress begins its exponential increase. The second parameter /3 is the most important one, and it indicates the rate of the exponential increase in stress with increasing strain. There have been only three (14, 20, 2 1) studies of the pericardial stress-strain relationship. The only previous mathematical model proposed for this relationship was an inverse parabola (14). TABLE
2. Effect of varying initial
II a
P x
S
0.000606 *o .ooo22 8.35 ho.42 0666 0
l
la
I2 4 I
-I
length on estimation of
I
0.000861 *o mo3 8.69 dIO.44 0.0665
l4 I
0 JO1225 &0.0004 9.03 zto.47 0 -0663
0.001747 &O .00058 9.36 ho.5 0.0661
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MATHEMATICAL
AND MECHANICAL PER ICARD
MODELING
PERICARDIUM
UM +COLlAGEN
SPRING
MYUCARDIUM
‘r
EIASTIN
SPRING
El FIG. 2. This is a diagramatic representation parie t al pericardium, and space between Parietal pericardium is represented by a collagen fibers (thicker springs) connected senting elastin fibers. (See text for further
of heart (open square), them (pericardial space). set of springs representing in parallel to a set repreexplanation of this model.)
Table 1 shows that an exponential expression is better than a parabolic one. Formulas for the precision of a and p estimates in equation 3 were derived here which can be used for statistical comparison of changes in the parameters following different interventions. There are other approaches in modeling the pericardial stress-strain relationship. The use of the natural strain definition (18) has advantages but requires a transformation which is not good for mathematical modeling. Fitting only one part of the stress range simplifies the problem but limits the validity of an equation in describing the stress-strain relationship. Redefining Poisson’s ratio (v) for the pericardium by estimating cy, & and Y simultaneously from the data may improve the fit for the medium strains. However, this would increase the complexity of the equation and the estimation of its parameters. This approach is of interest because equations 3 and 5 are equivalent when Y = 0. The usefulness of a model to describe a system has been clearly stated by Berman (3). The following mechanical model was formulated for the parietal pericardium. Two sets of springs, one representing the collagen fibers and one representing the elastin fibers of the pericardium, are connected in parallel (see Fig. 2); the collagen springs are set initially at less than their unstretched length. This spring nest (combination of springs) is stabilized by being fixed to surrounding mediastinal structures and great vessels by collagen springs which represent the extension of collagen fibers into the major cardiac vessels and into the ligaments to other mediastinal structures (8). The springs which are compressed or extended are those oriented so that all or part of the load is applied to their helical axis. The representation of the collagen and elastin fibers by two springs in parallel and one dashpot allows the model to describe not only the stressstrain relationship, but also the decline in tension seen during stress relaxation (10, 21). The proposal that the -collagen spring be at an unstretched length is consistent with microscopic examination of collagen fibers subjected to strain. Increasing strain produces increasing straightening and orientation of collagen fibers in the direction of the load (17). After the collagen fibers have undergone a change in geometric configuration, they begin to carry a greater proportion of the load. This is also consistent with the finding that the parameter a in equation 3 is larger in pericardium consisting of collagen tissue (after elastic tissue digestion). It is no doubt naive to view the pericardium as composed of two sets of springs in parallel. Other cellular and noncellular elements of the pericardium may contribute to the pericardium’s mechanical properties. Furthermore, the elastin and collagen fibers are probably not simple separate
899
springs but are interrelated in a complex manner. This model is proposed because springs are commonplace objects and can be readily conceptualized. Second, the mechanical properties of the pericardium are similar to those of a spring. A spring is a mechanical body whose primary function is to deflect or distort under a load and in which the amount of deflection is closely related to the applied load (4). The extrapolation of data from uniaxial loading to three-dimensional stress has ample precedents in the studies of myocardium (15, 16, 18). The elastic part of the pericardial model, which is easier to characterize, can be simulated by the equation c = a[@’ - I]. A viscoelastic element was included in the pericardial model because the pericardium has viscoelastic properties (Z), and it can aid in describing the role of the pericardium in disease. The concept of the pericardium represented by springs is useful because it can describe the behavior of the pericardium under different stimuli. Several examples are as follows. The proposed role of the pericardium in the prevention of overdilatation of the heart (1) can be visualized. Dilatation of the heart compresses the springs which exert a force against the heart limiting its dilatation. In pericardial effusion the accumulation of fluid compresses the springs (Fig. 3). Small volumes result in slight compression of primarily the elastin springs as the collagen springs are set at slightly more than their stretched length. As more fluid compresses the spring, the rising pericardial pressure accelerates (19) This is because the stiffer collagen springs become compressed and the nature of both springs is exponential. The same pericardial fluid volume in the hypervolemic state results in greater pericardial pressure than in the normovolemic one (19) because in hypervolemia the volume of pericardial fluid is applied onto the springs already compressed by an enlarged heart. The converse is found in hypovolemia. In constrictive pericarditis the pericardial space has been obliterated by scar tissue which is also attached to the myocardium. Systole and its resultant decrease in myocardial size extends the pericardial springs. The force exerted by the pericardial springs is in the oppo-
MYOCAR
D I UM
2 3m t E d
L VOLUME
FIG. 3. Crpp,, fianel: on left a model of pericardial effusion which has a volume sufficient only to compress elastin spring and on right resultant pressure volume curve of pericardium. Lower #and: on left a model of pericardial effusion with a volume sufficient to compress both springs of pericardium and on right resultant pressure volume curve.
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900 site direction and restrains myocardial fibers shortening. This may explain the depressed left ventricular contractility in constrictive pericarditis (23). The limitation of diastolic expansion of the myocardium by the pericardial springs is responsible for the diminished end diastolic volume in cardiac tamponade and constrictive pericarditis (23). The higher filling pressure in constrictive pericarditis and cardiac tamponade reflects the higher pressure needed to overcome the resistance of the springs in order to fill the ventricle. This mechanical model can also provide insight into the behavior of the pericardium with possible explanations for previously unexplained observations. Three examples are as follows : despite the large pericardial pressure in experithe dog’s left atria1 lumen mental cardiac tamponade, always remains patent (5). In the proposed model, the relatively indistensible co11 .agen springs which are att ached to the pulmonary veins as they enter the atrium resist the pressure to collapse the veins. Also the pericardial fluid distends these collagen springs pull .ing the veins apart, thus opposing the pericardial fluid pressure which is trying to collapse them. A rapid accumulation of pericardial fluid can produce cardiac tamponade with smaller than usual pericardial fluid volume. This may be explained by two fa .cts, first the force exerted by a viscoelastic spring is proportional to the
S. W.
RABKIN
AND
P. H.
HSU
velocity of the mass which deflects the spring (4). Second, the rapid accumulation of pericardial fluid does not allow time for the stress-relaxation properties of the pericardial springs to be utilized. Rapid enlargement of the size of the heart in acute mitral regurgitation and acute ventricular septal rupture can similarly result in a large counterforce applied to the heart by the pericardial springs with resultant signs of cardiac compression (2, 9). The early diastolic dip in ventricular pressure may be explained by a suction effect created because the collagen springs which have been extended by systole rapidly pull the myocardium back to the springs’ resting position. (A similar explanation was proposed by I3urch and Giles (6)) The rapid rise of ventricular pressure in diastole to a plateau (22) may be analogous to the exponential rise in resistance during compression of the pericardial springs by the enlarging left ventricle. The plateau may be analogous to the stress-relaxation relationship of the pericardium (21) associated with a small continued loading of the springs from the small increase in ventricular volume in late diastole. We are grateful to Dr. R. C. Shlant the development of this work.
for his invaluable
Received
1974.
for publication
12 December
assistance
in
REFEIZENCES 1. BARNARD, H. L. The function of the pericardium. Proceedings of the Physiological Society. J. Physiol., London 22 : 43-48, 1898. 2. BARTLE, S. H., AND H. J. HERMAN. Acute mitral regurgitation in man : hemodynamic evidence and observations indicating an early role for the pericardium. Circulation 36: 839-846, 1967. 3. BERMAN, M, The application of multicompartmental analysis to the problems of clinical medicine. Ann. Internal Med. 68: 424-428, 1968. 4. BERRY, W. R. Spring Design. A Practical Treatment. London: Emmott, 1965, p, 9-l 9. 5. BRECHER, G. A., AND C. A. GILBERT. Atria1 transmural pressure during experimental pericardial tamponade. Circulation Res. 28 : 323-329, 1971. 6. BURGH, G. E., AND T. D. GILES. Theoretic considerations of the “dip” of constrictive pericarditis. (Annotation). Am. post-systolic Heart J. 86: 569, 1973. 7. DRAPER, N., AND H. SMITH. Applied Regression Analysis. New York : Wiley, 1966, p. 263-30 1. 8. ELIAS, II,, AND L. J. BOYD. Notes on the anatomy, embryology and histology of the pericardium. J. Iv.Y, Med. Coil. 2: 50-75, 1960. 9. FEEST, T. G., G. C. SUTTON, R. J. VECHT, AND R. V. GIBSON. Signs of pericardial constriction in rupture of ventricular septum complications myocardial infarction. Bit. Heart J. 34: 1176-l 180, 1972. 10. FUNG, Y. C. Foundations of Solid Mechanics, Englewood Cliffs, N. J. Prentice, 1965, p* 20-25. 11. FUNG, Y. C. Elasticity of soft tissue in simple elongation. Am. J. Physiol. 2 13 : 1532-l 544, 1967. 12. FUNG, Y. C. B. Stress-strain-history relations of soft tissues in simple elongation. In : Biomechanics-Its Foundation and Objectives, edited by Y. C. Fung, N, Perrone, and M. Anliker. Englewood Cliffs, N. J. : Prentice, 1972, p. 181-208. 13. GUYTON, A. C., T. G. COLEMAN, AND H.J. GRANGER. Circulation: overall regulation. Ann. Rev. Physiol. 34: 13-51, 1972.
14. HILDEBRANDT, J., H. FUKAYA, AND C.J. MARTIN. Simple uniaxial and uniform biaxial deformation of nearly isotropic incompressible tissues. Biophys. J. 9: 781-791, 1969. 15. HUGENHOLTZ, P. G., R. C. ELLISON, C. W. URSCHEL, I. MIRSKY, AND E. H. SONNENBLICK. Myocardial force-velocity relationships in clinical heart disease. Circulation 41 : 191-202, 1970. 16. JANZ, R. F., AND A. F. GRIMM. Deformation of the diastolic left ventricle. I. Nonlinear elastic effects. Biofhys. J. 158: 689-704, 1973. 17. KENEDI, R. M., T. GIBSON, AND C. I-3. DALY. Bioengineering studies of the human skin, the effect of unidirectional tension. In: Structure and Fumtion of Connective and Skeletal Tissue, edited by S. F. Jackson, S. M. Harkness, and G. R. Tristram. St. Andrews, Scotland : Scientific Committee, 1964, p* 388-395. 18. MIRSKY, I., AND W. W. PARMLEY. Assessment of passive elastic stiffness for isolated heart muscle and the intact heart. Circulation Res. 33: 233-243, 1973. 19. MORGAN, B. C., W. G. GUNTHEROTH, AND D, H. DILLARD. Relationship of pericardial to pleural pressure during quiet respiration and cardiac tamponade. Circulation Res. 16 : 493-498, 1965. 20. NELEMANS, F. A. Die Funktion des Perikards. Arch. Neerl. Physiol. 24: 337-390, 1939-1940. 21. RABKIN, S. Mr., D. G. BERGHAUSE, AND H. F. BAITER. Mechanical properties of the isolated canine pericardium. J. A$@. Physiol. 36: 69-73, 1974. 22. ROACH, M. E., AND A. C. BURTON. The reason for the shape of the distensibility curves of arteries. Can. J. Biochem. Physiol. 35: 681-690, 1957. 23. SHABETAI, R. N., 0. FOWLER, AND W. G. GUNTHEROTH. The hemodynamics of cardiac tamponade and constrictive pericarditis. Am, J. Cardiol. 26: 480-489, 1970. 24. SONNENBICK, E. H. Series elastic and contractile elements in heart muscle : change in muscle length. Am. J. Physiol. 207: 1330-l 338, 1962.
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JOURNAL
OF PHYSIOLOGY
Vol. 229, No. 4, October
1975.
Prinkdin
U.S.A.
Mathematical
and mechanical
of stress-strain
relationship
modeling
of pericardium
SIMON W. RABKIN, AND PING HWA HSU Department of Medicine, Sectiolz of Cardiology, and De@rtment and Epidemiology, University of Manitoba, Winnipeg, Canada
W., AND PING HWA Hsu. Mathematical and mestress-strain relationship of pericardium. Am. J. Physiol. : 896-900. 1975.-Several mathematical expressions (models) were compared for use in describing the stressstrain (a - C) relationship of pericardium. The expression c = a[eBc - I] was preferred because of its simpler form, theoretical consistency, and “good fit” of experimental data. A method was developed for estimating the precisions of the estimates of the parameters cy and & This approach can have general usefulness in assessing the significance of a change in stress-strain relationship of various soft tissues following different interventions. A mechanical model was formulated for the pericardium which consisted of springs representing the collagen and elastin fibers connected in parallel. It could be simulated by the above equation and could describe the behavior of the pericardium. RABKIN,
chanical
SIMON
modeling 229(4)
biomechanics;
of
statistical
analysis;
spring
FORCE-DEFORMATION
(stress-strain)
METHOD
The numerical from pericardial
data for this analysis were taken manually force-deformation curves which had formed
Preventive Medicine,
Section
of Biostati&s
the basis of a previous report (2 1). The first point was the one at the onset of deformation, and the last point was the one just prior to specimen failure. In between and including these two points, a total of 25 were transcribed. Six intact pericardial specimens and three specimens subjected to elastic tissue digestion were characterized (see Table 1 in a previous report) (2 1). Stress ((T) is defined as force per unit area, i.e., crosssectional area determined prior to deformation and is expressed as kilograms per square millimeter. Strain (E) is defined as the length changes per unit initial length; to the onset (1 - LJ/lo 3 where 1, is the length corresponding of deformation and 1 is the instantaneous length. Five equations derived for other soft tissues are: u = PE2
(I)
u = aE + pE2
(2)
model
relationship is a fundamental property of living tissue (11, 12, 14, 17, 18). Rabkin et al. (21) recently studied this property in pericardium and observed that it had the same nonlinear stressstrain relationship as other soft tissues. This nonlinear relationship would be best described by a mathematical expression because it would provide a concise description, facilitate data collection, and assist further quantitative research (11). In this report several mathematical expressions proposed for the elastic properties of other soft tissue (11, 12, 16-18) were compared for the pericardium, and one expression was selected and justified. For the purpose of comparison, the precision of the estimates of the parameters in the equation was also derived. The formulation of a model to describe a system is also a useful approach because a model can be used to concisely describe the behavior of the system, to gain insight into its behavior under different stimuli, and to predict its behavior (3). Models have been proposed for many aspects of the cardiovascular system (13, 24), but none for the pericardium. A mechanical model will be proposed here in the hope that it can descri be some of its functions and explain some of its d *isorders. THE
of Socialand
u=
a[e BE -
11 (W
(3)
u=
aeBt + y
(18)
(4)
u
=
[a/@(
1
+
v)]
n
[ep’
-
e-+‘] (5)
Equation 5 is modified from Janz and Grim (16) wherein v is Poisson’s ratio (= 0.49 for incompressible tissues) and a is now a parameter to be estimated. Two other equations were also compared : do/de
= a + flu
(11)
do/de
= Ql(l + pa>,
(6) (7)
Equation 3 can be derived from equation 6 as follows: the extension ratio (X) was defined as the length of the tissue under strain divided by the resting length of the tissue (l/lo). From experimental results, it has been observed that the tangent (da/dX) is correlated with the stress (c) (11). As a first approximation, it can be fitted by a straight line: da/dX = P(a + cu)* An integration gives c + a = (l/c)#” with an integration constant c. If u = C* when X = h”, then the equation becomes g = (g* + a) $(h-‘*) - a. By definiwith tion, g = 0 when X = 1. If C* and X* are substituted these theoretical values, the equation reduces to : a($@ - I) - 1) or fl = a(eSE - 1) with E = X - 1 = ;; =_ lo& Th’ IS is equation 3. Note equation 3 is equivalent to equation 5 when u = 0. Each mathematical expression (e.g., Eq. 1-7) was placed in the form Y = f(X, 0) + e, where Y = stress vector, X =
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MATHEMATICAL
AND
MECHANICAL
MODELING
strain vector, 8 = parameter vector which characterized the mathematical expression, and c is the error vector. The least-square method was used to estimate 0, i.e., choosing a 0 vector such that the sum of squares of deviations, e’e, is a minimum. The normal equations were derived by differentiating the equation for e’e with respect to 8, and then the equations were set equal to zero. The estimators of 8 were then solved. For the linear equations the solution was straightforward, but for the nonlinear equations iterative schemes derived from the normal equations were employed (7). The residual sum of squares for each mathematical expression was calculated for comparison of the goodness of A scheme of linearization, together the precision formuIa for the estimates, for equation 3. Equation 3 may be rewritten in the Xi = Ei,yi = a (esxi - 1) $- Ei. This be linearized by a Taylor’s expansion Q! and fl(a~(*), PC”‘) as follows : Fi(*) = + Ei where
with the derivation of is outlined as follows form: for y i = pi and nonlinear model may around trial values of &(*)ZJ”) + p2’O)Zi2’0)
&(*I pp>
= y i - ab~(e&*)zi = a - &I Pz(*) z p - pb> Zil(O’ = &(O)Xi - 1 Zi2(0)
=
897
PERICARDIUM COmpariSOn of SeVera! mathematical stress-strain rdationship
1.
TABLE
pericardial I
z If
% + ii II
-rl I 5 T II
5 II
b
b
b
b
--
-~
a
B
-1.26 0.83
2.46
1.018
0.45
a B i s c
--.393 7.05
2.73
16.9
0.96 -1.80
aC*)xi,jj(obi
is E
lzs
g42 E
Eb
i-
These precision estimates are calculated after the estimates of it and p are obtained by iteration. These estimates are small and within the linearized range (as will be seen later). A simpler method for estimation of a and P in equation 3 was derived as follows. The equation S (a, P) = 2[JJiQ(Efixi - l)]” is differentiated with respect to cy and P* Setting the results equal to zero gives two normal equations:
cy= c YiP - 1) - 1)” CCeSxi C C
XiPi($ri
#iyiP' -
1)
Since a should be equal in the above two equations, the iteration scheme can be designed to choose P such that these two equations are nearly equal. The value which is equal in both equations is cy. Both these methods should give similar results, but the latter method doesn’t give the precision
Cs - -~
0.00818 AO.0037 11.78 +0.98 0.5208 0.01865 +O. 0062 5.10 dzo.37
2.06
0.45
0.4389
0.94
0.0020 d~O.00081 5.19 zto.33
0.55
1.45
1.30
0.47 -0.49
0.40
1.05
0.22
0.095 -1.63
1.96
5.1
with l)]”
0.0666
4.20
a
B
0.00062 zto. 00022 8.35 ~0.42
1.90
a
F
&(eSxi n-2
for
b
A
- 1)
Since this nonlinear equation is linearized, an approximate precision of the estimator fi and $i can be derived from least-squares theory. They are
models
j 0.84
0.28
0.1392 0.00038 *0.00011 7.96 ho.32 0.00993 0.00212 +O .00065 10.04 hO.50 0.05740
0.00085
-0.0216 0.0712 0.01226
0.00669
0.5042
t4.88
8.40
5.34
0.65
46.13
1.79
0.1005 0.1308
10.95 -0.0778 0.4726
4.55 -0.1271 0.3804
0.560
5.20
0.5860
1.1257
!5.98
7.12
0.370
Z81.18
0.41
0.7326
0.09
3.11
0.368
45.65
3.769
0.00294
0.0135
0.2901
7.861
4.85
5.20
3.20
0.414
18.007
2.06
-0.0392 0.1412
0.2155 0.00403
0.0594
7.847
8.0
6.35
0,029
-0.00185 0.01181
0.0172
7.23
4.148
0.00306
0.0290
0.4482
15.79
6.82
0.559
20.18
0.865
7.85
9.45 -0.03154 0.05129
10.10
0.0734
estimate. One may estimate ct! and @ by the latter and use these estimates in the precision formula from the former method.
method derived
RESULTS
Goodness of jt. In Table 1 the estimated values of the parameters a, ,6, y, and the residual sums of squares corresponding to each mathematical expression are listed for six intact pericardia. Over the stress range O-2.5 kg/mm2, which corresponds to a range of length change from 0 to 128 % of resting length (lo), the two polynomial equations were consistently poorer in fit compared to the three exponential functions. The goodness of fit for the exponential functions is demonstrated in Fig. 1 where specimen A is displayed along with equations 3 and 5. Equation 4 was not included because it predicts negative stress for small strains. All three exponential functions have similar residual sums of squares and give similar estimates of the most important parameter, /3. If a choice must be made among these three expressions, equation 4 will be rejected because it is inconsis-
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898
S. W.
-
ACTUAL
0.9 -
Y
0.6 -
% g os+ UJ 0.4 0.3 -
STRAIN FIG. 1. A typical stress-strain curve for a specimen of canine pericardium is shown in solid line. Two broken lines represent 2 exponential equations 3 and 5 which best fit actual data.
tent with a theoretical consideration (0 = 0 when e = 0), it predicts negative stress when strain is small (G < 0 when E = 0), and y is always negative and larger than P. Equation 5 will be rejected because of its complex form and difficulty in the estimation of the parameter p when Q is not estimated simultaneously. Equation 5 may be more useful in multidimensional problems (16). Equation 3 was selected to best represent pericardial stress-strain relationship. As can be seen, it is simple in form, theoretically consistent, a good “fit” to the experimental by only d ata, and is characterized two par ameters. For these reasons it was the only one subjected to further analysis. The comparison of the “goodness of fit” between the first-order (equation 6) and second-order (equation 7) approximation indicated that the second-order approximation does fit better. However, integration of equation 7 gives a complicated formula which is no better than equation 3 and is also rejected at this time. Effect of selection of 8 and A*. In the derivation of equation 3 (see METHOD), U* = 0 and X* = 1 were chosen because of theoretical considerations. Since G* is the stress at a specific extension rate X”, one might choose any point (G*, X*) on the stress-strain curve. For every pericardial specimen, the choice of a point other than (0, 1) results in equations predicting negative stress. This is inconsistent with the stressstrain definition and therefore another reason to define strain as the length change per unit initial length. Determination of I, . The determination of a specimen’s resting length (lo) is difficult. The extent to which variations in this measurement could affect equation 3 was analyzed. Table 2 gives four estimates of cy, /3, and residual sum of squares obtained by deleting none, the first one, first two, and first three data points, i.e., simulation of the situations that lo, II, 12, and 13 are used in place of I,. For each consecu-
RABKIN
AND
P. H. HSU
thin 1 SE tive change there is an increase in /3 which is and an increase 1n QTwhich is more marked. .e resid ual sum of squares declines which is expected because a unimodal equation is being used to fit data which are essentially bimodal. The bias in each consecutive step is about 1 mm. These results are similar in all other sets of data and indicate that roughly measured I, does not affect the estimation of the most important parameter P by more than 1 SE which here is a small one. The average of P in these six specimens is 8.08, compared with the average Young’s modulus of 6.15 kg/mm2 (21). The similar order of magnitude suggests that an estimate of Young’s modulus can be used as an initial value in the iterative estimation of /3 for convenience. Efect of elastic tz’ssue digestion, The three pericardia subjected to elastic tissue digestion had the following parameters of a and p estimated for equation 3: 0.271 X 10m4A 0.101 X lO-4 (+ 1 SE) and 11.24 j= 0.54; 0,021.l =t 0.0059 and 1.87 & 0,15; 0.2242 =t 0.0594 and 4.47 & 0.66. When they are compared to the same specimen not subjected to elastic tissue digestion (B, D, E, in Table 1, respectively), the last two sets show significant changes in fl (in l-tests, P < 0.01). The larger ct after elastic tissue digestion shows a change not apparent on initial inspection of the force deformation curves, This change is similar to that found by Roach and Burton (22) in arteries subjected to elastic tissue digestion. DISCUSSION The necessity to utilize a mathematical expression (model) for the stress-strain relationship of soft tissues has been clearly stated by Fung (1 l)* He has also pointed out (12) that the ultimate mathematical model is still a distant goal. The elastic properties of the pericardium, not the viscoelastic ones, were herein subjected to mathematical modeling because they a.re more easily studied and conceptualized. It has been generally accepted that a good model should be simple, characterized by few parameters and uncomplicated in form; realistic, consistent with theory; and accurate, fits the experimental data well and precisely estimates the parameters. The equation 0 = a[efi” - l] was chosen because it is relatively simple in form, characterized by only two parameters and consistent with the theoretical consideration that when e = 0, u = 0. The first parameter ~1 characterizes the level of strain at which stress begins its exponential increase. The second parameter /3 is the most important one, and it indicates the rate of the exponential increase in stress with increasing strain. There have been only three (14, 20, 2 1) studies of the pericardial stress-strain relationship. The only previous mathematical model proposed for this relationship was an inverse parabola (14). TABLE
2. Effect of varying initial
II a
P x
S
0.000606 *o .ooo22 8.35 ho.42 0666 0
l
la
I2 4 I
-I
length on estimation of
I
0.000861 *o mo3 8.69 dIO.44 0.0665
l4 I
0 JO1225 &0.0004 9.03 zto.47 0 -0663
0.001747 &O .00058 9.36 ho.5 0.0661
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MATHEMATICAL
AND MECHANICAL PER ICARD
MODELING
PERICARDIUM
UM +COLlAGEN
SPRING
MYUCARDIUM
‘r
EIASTIN
SPRING
El FIG. 2. This is a diagramatic representation parie t al pericardium, and space between Parietal pericardium is represented by a collagen fibers (thicker springs) connected senting elastin fibers. (See text for further
of heart (open square), them (pericardial space). set of springs representing in parallel to a set repreexplanation of this model.)
Table 1 shows that an exponential expression is better than a parabolic one. Formulas for the precision of a and p estimates in equation 3 were derived here which can be used for statistical comparison of changes in the parameters following different interventions. There are other approaches in modeling the pericardial stress-strain relationship. The use of the natural strain definition (18) has advantages but requires a transformation which is not good for mathematical modeling. Fitting only one part of the stress range simplifies the problem but limits the validity of an equation in describing the stress-strain relationship. Redefining Poisson’s ratio (v) for the pericardium by estimating cy, & and Y simultaneously from the data may improve the fit for the medium strains. However, this would increase the complexity of the equation and the estimation of its parameters. This approach is of interest because equations 3 and 5 are equivalent when Y = 0. The usefulness of a model to describe a system has been clearly stated by Berman (3). The following mechanical model was formulated for the parietal pericardium. Two sets of springs, one representing the collagen fibers and one representing the elastin fibers of the pericardium, are connected in parallel (see Fig. 2); the collagen springs are set initially at less than their unstretched length. This spring nest (combination of springs) is stabilized by being fixed to surrounding mediastinal structures and great vessels by collagen springs which represent the extension of collagen fibers into the major cardiac vessels and into the ligaments to other mediastinal structures (8). The springs which are compressed or extended are those oriented so that all or part of the load is applied to their helical axis. The representation of the collagen and elastin fibers by two springs in parallel and one dashpot allows the model to describe not only the stressstrain relationship, but also the decline in tension seen during stress relaxation (10, 21). The proposal that the -collagen spring be at an unstretched length is consistent with microscopic examination of collagen fibers subjected to strain. Increasing strain produces increasing straightening and orientation of collagen fibers in the direction of the load (17). After the collagen fibers have undergone a change in geometric configuration, they begin to carry a greater proportion of the load. This is also consistent with the finding that the parameter a in equation 3 is larger in pericardium consisting of collagen tissue (after elastic tissue digestion). It is no doubt naive to view the pericardium as composed of two sets of springs in parallel. Other cellular and noncellular elements of the pericardium may contribute to the pericardium’s mechanical properties. Furthermore, the elastin and collagen fibers are probably not simple separate
899
springs but are interrelated in a complex manner. This model is proposed because springs are commonplace objects and can be readily conceptualized. Second, the mechanical properties of the pericardium are similar to those of a spring. A spring is a mechanical body whose primary function is to deflect or distort under a load and in which the amount of deflection is closely related to the applied load (4). The extrapolation of data from uniaxial loading to three-dimensional stress has ample precedents in the studies of myocardium (15, 16, 18). The elastic part of the pericardial model, which is easier to characterize, can be simulated by the equation c = a[@’ - I]. A viscoelastic element was included in the pericardial model because the pericardium has viscoelastic properties (Z), and it can aid in describing the role of the pericardium in disease. The concept of the pericardium represented by springs is useful because it can describe the behavior of the pericardium under different stimuli. Several examples are as follows. The proposed role of the pericardium in the prevention of overdilatation of the heart (1) can be visualized. Dilatation of the heart compresses the springs which exert a force against the heart limiting its dilatation. In pericardial effusion the accumulation of fluid compresses the springs (Fig. 3). Small volumes result in slight compression of primarily the elastin springs as the collagen springs are set at slightly more than their stretched length. As more fluid compresses the spring, the rising pericardial pressure accelerates (19) This is because the stiffer collagen springs become compressed and the nature of both springs is exponential. The same pericardial fluid volume in the hypervolemic state results in greater pericardial pressure than in the normovolemic one (19) because in hypervolemia the volume of pericardial fluid is applied onto the springs already compressed by an enlarged heart. The converse is found in hypovolemia. In constrictive pericarditis the pericardial space has been obliterated by scar tissue which is also attached to the myocardium. Systole and its resultant decrease in myocardial size extends the pericardial springs. The force exerted by the pericardial springs is in the oppo-
MYOCAR
D I UM
2 3m t E d
L VOLUME
FIG. 3. Crpp,, fianel: on left a model of pericardial effusion which has a volume sufficient only to compress elastin spring and on right resultant pressure volume curve of pericardium. Lower #and: on left a model of pericardial effusion with a volume sufficient to compress both springs of pericardium and on right resultant pressure volume curve.
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900 site direction and restrains myocardial fibers shortening. This may explain the depressed left ventricular contractility in constrictive pericarditis (23). The limitation of diastolic expansion of the myocardium by the pericardial springs is responsible for the diminished end diastolic volume in cardiac tamponade and constrictive pericarditis (23). The higher filling pressure in constrictive pericarditis and cardiac tamponade reflects the higher pressure needed to overcome the resistance of the springs in order to fill the ventricle. This mechanical model can also provide insight into the behavior of the pericardium with possible explanations for previously unexplained observations. Three examples are as follows : despite the large pericardial pressure in experithe dog’s left atria1 lumen mental cardiac tamponade, always remains patent (5). In the proposed model, the relatively indistensible co11 .agen springs which are att ached to the pulmonary veins as they enter the atrium resist the pressure to collapse the veins. Also the pericardial fluid distends these collagen springs pull .ing the veins apart, thus opposing the pericardial fluid pressure which is trying to collapse them. A rapid accumulation of pericardial fluid can produce cardiac tamponade with smaller than usual pericardial fluid volume. This may be explained by two fa .cts, first the force exerted by a viscoelastic spring is proportional to the
S. W.
RABKIN
AND
P. H.
HSU
velocity of the mass which deflects the spring (4). Second, the rapid accumulation of pericardial fluid does not allow time for the stress-relaxation properties of the pericardial springs to be utilized. Rapid enlargement of the size of the heart in acute mitral regurgitation and acute ventricular septal rupture can similarly result in a large counterforce applied to the heart by the pericardial springs with resultant signs of cardiac compression (2, 9). The early diastolic dip in ventricular pressure may be explained by a suction effect created because the collagen springs which have been extended by systole rapidly pull the myocardium back to the springs’ resting position. (A similar explanation was proposed by I3urch and Giles (6)) The rapid rise of ventricular pressure in diastole to a plateau (22) may be analogous to the exponential rise in resistance during compression of the pericardial springs by the enlarging left ventricle. The plateau may be analogous to the stress-relaxation relationship of the pericardium (21) associated with a small continued loading of the springs from the small increase in ventricular volume in late diastole. We are grateful to Dr. R. C. Shlant the development of this work.
for his invaluable
Received
1974.
for publication
12 December
assistance
in
REFEIZENCES 1. BARNARD, H. L. The function of the pericardium. Proceedings of the Physiological Society. J. Physiol., London 22 : 43-48, 1898. 2. BARTLE, S. H., AND H. J. HERMAN. Acute mitral regurgitation in man : hemodynamic evidence and observations indicating an early role for the pericardium. Circulation 36: 839-846, 1967. 3. BERMAN, M, The application of multicompartmental analysis to the problems of clinical medicine. Ann. Internal Med. 68: 424-428, 1968. 4. BERRY, W. R. Spring Design. A Practical Treatment. London: Emmott, 1965, p, 9-l 9. 5. BRECHER, G. A., AND C. A. GILBERT. Atria1 transmural pressure during experimental pericardial tamponade. Circulation Res. 28 : 323-329, 1971. 6. BURGH, G. E., AND T. D. GILES. Theoretic considerations of the “dip” of constrictive pericarditis. (Annotation). Am. post-systolic Heart J. 86: 569, 1973. 7. DRAPER, N., AND H. SMITH. Applied Regression Analysis. New York : Wiley, 1966, p. 263-30 1. 8. ELIAS, II,, AND L. J. BOYD. Notes on the anatomy, embryology and histology of the pericardium. J. Iv.Y, Med. Coil. 2: 50-75, 1960. 9. FEEST, T. G., G. C. SUTTON, R. J. VECHT, AND R. V. GIBSON. Signs of pericardial constriction in rupture of ventricular septum complications myocardial infarction. Bit. Heart J. 34: 1176-l 180, 1972. 10. FUNG, Y. C. Foundations of Solid Mechanics, Englewood Cliffs, N. J. Prentice, 1965, p* 20-25. 11. FUNG, Y. C. Elasticity of soft tissue in simple elongation. Am. J. Physiol. 2 13 : 1532-l 544, 1967. 12. FUNG, Y. C. B. Stress-strain-history relations of soft tissues in simple elongation. In : Biomechanics-Its Foundation and Objectives, edited by Y. C. Fung, N, Perrone, and M. Anliker. Englewood Cliffs, N. J. : Prentice, 1972, p. 181-208. 13. GUYTON, A. C., T. G. COLEMAN, AND H.J. GRANGER. Circulation: overall regulation. Ann. Rev. Physiol. 34: 13-51, 1972.
14. HILDEBRANDT, J., H. FUKAYA, AND C.J. MARTIN. Simple uniaxial and uniform biaxial deformation of nearly isotropic incompressible tissues. Biophys. J. 9: 781-791, 1969. 15. HUGENHOLTZ, P. G., R. C. ELLISON, C. W. URSCHEL, I. MIRSKY, AND E. H. SONNENBLICK. Myocardial force-velocity relationships in clinical heart disease. Circulation 41 : 191-202, 1970. 16. JANZ, R. F., AND A. F. GRIMM. Deformation of the diastolic left ventricle. I. Nonlinear elastic effects. Biofhys. J. 158: 689-704, 1973. 17. KENEDI, R. M., T. GIBSON, AND C. I-3. DALY. Bioengineering studies of the human skin, the effect of unidirectional tension. In: Structure and Fumtion of Connective and Skeletal Tissue, edited by S. F. Jackson, S. M. Harkness, and G. R. Tristram. St. Andrews, Scotland : Scientific Committee, 1964, p* 388-395. 18. MIRSKY, I., AND W. W. PARMLEY. Assessment of passive elastic stiffness for isolated heart muscle and the intact heart. Circulation Res. 33: 233-243, 1973. 19. MORGAN, B. C., W. G. GUNTHEROTH, AND D, H. DILLARD. Relationship of pericardial to pleural pressure during quiet respiration and cardiac tamponade. Circulation Res. 16 : 493-498, 1965. 20. NELEMANS, F. A. Die Funktion des Perikards. Arch. Neerl. Physiol. 24: 337-390, 1939-1940. 21. RABKIN, S. Mr., D. G. BERGHAUSE, AND H. F. BAITER. Mechanical properties of the isolated canine pericardium. J. A$@. Physiol. 36: 69-73, 1974. 22. ROACH, M. E., AND A. C. BURTON. The reason for the shape of the distensibility curves of arteries. Can. J. Biochem. Physiol. 35: 681-690, 1957. 23. SHABETAI, R. N., 0. FOWLER, AND W. G. GUNTHEROTH. The hemodynamics of cardiac tamponade and constrictive pericarditis. Am, J. Cardiol. 26: 480-489, 1970. 24. SONNENBICK, E. H. Series elastic and contractile elements in heart muscle : change in muscle length. Am. J. Physiol. 207: 1330-l 338, 1962.
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