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Regional mechanical properties of passive myocardium
J. Biomechanics, Vol. 21, No. 4, pp. 40-412,
Pergamon
1994
Copyright 0 1994 Ekvier Science Ltd Printed in Great Britain. All rights reserved OoZl -9290/94 $6.00 + .@I
REGIONAL
MECHANICAL PROPERTIES MYOCARDIUM
OF PASSIVE
V. P. NOVAK, F. C. P. YIN* and J. D. HUMPHREYP Department of Mechanical Engineering, The University of Maryland, Baltimore, MD 21228, U.S.A. and *Departments of Medicine and Biomedical Engineering, The Johns Hopkins Medical Institutions, Baltimore, MD 21205, U.S.A. Abstrati-There is considerable interest in calculating regional stresses in the heart. This, in turn, requires complete information on regional material properties which, surprisingly, is not available. The specific aim of this work, therefore, was to determine if transmural differences exist in the mechanical behavior of passive myocardium in the equatorial region of the heart. Thus, we performed in uitro biaxial experiments on 28 thin, rectangular slabs of noncontracting myocardium excised from four regions within canine hearts: the middle portion of the interventricular septum (n = 8), and the inner (n = 5), middle (n = 9) and outer (n = 6) layers of the lateral left ventricular (LV) free wall. There were three major findings. First, an existing threedimensional constitutive relation described the nonlinear and anisotropic behavior exhibited in the four regions equally well. Second, the anisotropy was similar in each region. Third, there were, however, regional differences in the strain-energy stored by specimens during identical finite deformations. In particular, specimens from inner and outer portions of the LV free wall tended to be stiffer than those from the middle of the LV free wall and septum. These findings, together with previous results on excised eoicardium. suggest that the mechanical uroDerties of the heart are qualitatively similar from region to rkgion, but quantitatively different.
INTRODUCTION A detailed understanding of cardiac physiology and pathophysiology requires quantification of regional phenomena, including distributions of stress and strain. Biplane cineradiography, one- and two-dimensional sonomicrometry, and most recently magnetic resonance imaging (MRI) provide important information on regional strains in the intact beating heart. In contrast, regional stresses cannot be measured reliably with the current methods. Rather, they must be calculated using the methods of continuum mechanics, which in turn requires knowledge of regional mechanical properties. The need for data on the regional biomechanical behavior of cardiac tissue has been known for many years, as has the potential importance of this information. For example, Mirsky (1970) showed that predicted transmural wall stresses differ significantly when myocardium is assumed to be heterogeneous rather than homogeneous as assumed by most. Similar findings were obtained by Janz and Grimm (1972, 1973) using element analyses of the left ventricle. Janz and Waldron (1976) and Moriarty (1980) later hypothesized that circumferential wall stress should be nearly uniform across the wall of the heart at end diastole, not much larger at the endocardium as predicted by most early analyses (Huisman et al., 1980). Using simplified models, these investigators showed that a uniform transmural stress distribution could exist in the heart if values of a circumferential modulus increased monotonically from the endocar-
dium to the epicardium. Although their specific results must be viewed cautiously-they neglected anisotropy due to transmural fiber distributions (Humphrey and Yin, 1989a), residual stress (Omens and Fung, 1990) and twist in the passive heart (Guccione et al., 1991), each of which significantly affects predicted transmural stress distributions-implications of heterogeneous wall properties were clearly demonstrated. Despite these provocative findings as well as direct calls for data (Huisman et al., 1980; Mirsky, 1979; Yin, 1981), there remains little information on the regional mechanical properties of cardiac tissue. The goal of this work, therefore, was to determine if transmural differences exist in the mechanical behavior of passive myocardium in the equatorial region of the canine heart. Consequently, (a) we performed in oitro biaxial experiments on noncontracting myocardium excised from four regions within canine hearts: the middle of the interventricular septum and inner, middle and outer portions of the lateral left ventricular (LV) free wall; (b) we quantified the observed biomechanical behavior using an existing three-dimensional nonlinear, anisotropic constitutive relation; (c) we compared the pseudostrain-energy stored, at identical deformations, by specimens from each of the four regions. This study provides, therefore, the first detailed information on the regional multiaxial stress-strain behavior of noncontracting myocardium. METHODS Specimen preparation
Received in final form 20 August 1993. t Author to whom correspondence should be addressed.
Mongrel dogs (u 20 kg) were anesthetized with sodium pentobarbital(30 mg kg- ‘), intubated and ven403
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The LV free wall and interventricular septum were then excised from the isolated heart, and the papillary muscles removed. Thin rectangular slabs ( N 2 cm x 2 cm x 0.25 cm) of myocardium were cut sequentially from both the lateral LV free wall and the septum, tangential to the epicardial or endocardial surfaces, using a rotary meat slicer. Although we obtained and tested multiple slabs from each heart, the onset of contracture prevented us from collecting ‘good’ data from a second specimen (i.e. region) in 14 of 21 hearts. Thus, we report data from 28 specimens taken from 21 hearts: 8 from the middle of the interventricular septum and 5, 9 and 6 specimens from inner, middle and outer portions of the lateral LV free wall, respectively. ‘Middle’ means within the middle-third of a wall, ‘inner LV’ corresponds to the subendocardium, just below the deepest invagination in the endocardial surface, and ‘outer LV specimens were taken N 1.5-2.0 mm below the epicardial surface.
to place specimens in the device such that the fiber splay (as judged by the eye) was symmetric with regard to the midplane; this reduces through-the-thickness variations in stresses (Humphrey and Yin, 1989b). Each specimen was coupled to the four carriages of the device using four continuous strands of 000 suture, each strand looped five times through one side of the specimen. Uniform spacing between each loop was ensured by sewing through a paper template that was placed atop the specimen; this template was removed prior to testing. A central region of the specimen was darkened with india ink, and four 1 mm diameter steel balls (4 mg each) were glued onto the specimen in a rectangular pattern within the central region; these balls demarcated -3% of the planar surface area. A white light source, directed at an angle to the surface of the specimen, produced a small bright spot on each ball that was tracked with custom video-based software (Downs et al., 1990). Once mounted, specimens were subjected to an initial preconditioning protocol that consisted of 7-10 cycles of equibiaxial loading (lo-45 g) at 0.1 Hz. These preconditioning protocols minimized nonrepeatable, history-dependent effects that result from excising, slicing and mounting a specimen. The unloaded, stress-free state of each specimen (i.e. outer dimensions and locations of the four tracking markers) was recorded following this preconditioning, and was checked three times to ensure repeatability. Each specimen was then subjected to five different cyclic stretching protocols conducted at 0.1 Hz: equibiaxial stretching (5-27%) and tests wherein the stretch in one direction was maintained constant at a prescribed value (‘v 10, 15 or 20%), while the orthogonal direction was cyclically stretched (5-27%). The equibiaxial protocol was performed first, and was repeated after the other protocols were finished. The first 5-7 cycles preconditioned the tissue for each specific protocol, whereas data were collected at 30 Hz from the last three cycles of each protocol. Specimen thickness was measured in the central region, at the completion of testing, using dial calipers. Specimens were then fixed in the unloaded state in a 10% buffered formalin, and stored at 5°C.
Biaxial tests
Fiber angles
Our test system and experimental methods are described elsewhere (Humphrey et al., 1990a, b; Yin et al., 1987). Briefly, a 386 PC controls two pairs of orthogonally mounted carriages via two independent sets of linear motors, worm gears and lead screws. A force transducer, riding on one carriage of each pair, measures the in-plane applied loads while a video camera, mounted above the specimen, monitors the deformation within a central region of the specimen. Resolution was 3 mN and 0.004 mm, respectively. Each rectangular specimen was immersed in the oxygenated cardioplegic solution at room temperature within the test chamber. In particular, we tried
Detailed information on orientations of muscle fibers in the canine heart is in Nielsen et al. (1991). Their data strongly suggest that for the regions studied herein, we could assume that the fiber splay was linear through the thickness of our thin (2-3 mm) slabs. Thus, we measured the predominant fiber angle on the top and bottom surfaces (QT and @a) in the central region of each fixed specimen. DT and @a were measured with a 10 x dissecting microscope and interactive software by recording coordinates along muscle fibers relative to orthogonal fiducial marks that were introduced in the specimen using a jig containing three heated 20 g needles. In addition, we measured through-the-thickness fiber distributions in
tilated. The chest was entered via a midline thoracotomy, and the pericardium opened. A 7F pigtail catheter was advanced from the right common carotid artery to just above the aortic valve. The subclavian and brachiocephalic arteries were ligated, the descending aorta cross-clamped, and the right atrium and left ventricle vented to atmosphere. The heart was then arrested in diastole with infusions of a cold heparinized (8000 units) oxygenated cardioplegic solution (in mM): 0.4 KH,PO*, 1.0 MgS04. 7H20, 28 NaHCO,, 1.5 CaClz. 2H,O, 5.6 dextrose, 117.5 L-glutamic acid and 1.0 adenosine which was used to dilate the coronary bed. Although 30 ml of solution typically induced cardiac arrest, 240-300 ml were infused to ensure thorough myocardial perfusion; return flow of solution was always evident in the epicardial coronary veins. The heart was then rapidly excised, and the circumflex, left anterior descending and septal coronary arteries perfused further with cardioplegic solution (N 120 ml). This resulted in an almost complete washout of blood from the myocardium. All animal care and use was IACUC approved, and thus consistent with the NIH Guide for the Care and Use of Laboratory Animals.
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Regional stresses in the heart
sequential 100 pm hematoxylin- and eosin-stained sections in 12 (three randomly selected from each region) of the 28 specimens, again using light microscopy. Data analysis
where El and Ez are orthonormal ing directions, and
bases in the stretch-
(5) H is the thickness of the undeformed
It is straightforward to calculate experimental biaxial stresses and strains from stress-free specimen dimensions, the two in-plane applied loads and the motions of four tracking markers (Humphrey et al., 1990a). Briefly, ‘measurable’ quantities of interest are the in-plane normal components (tll, tzz) of the Cauchy stress t and the in-plane components (&, rci, &, rc2) of the deformation gradient F. Subscripts 1 and 2 denote in-plane biaxial stretching directions (i.e. predominant fiber and cross-fiber directions, respectively). In the figures below, however, we plot Cauchy stresses vs the stretch ratios in the 1 and 2 directions:* Ai =J_,
A2 = dm.
(1)
To quantify the behavior of any material requires identification of a constitutive relation. For materials like passive myocardium, which exhibit complex nonlinear anisotropic behavior under finite strains, constitutive relations are often best posed in terms of a pseudostrain energy W. That is, a scalar function that represents the energy stored in a material during deformation. We used a W that we previously identified from biaxial tests on noncontracting canine myocardium excised from the midwall of the LV free wall (Humphrey et al., 1990b):
+c.+(11-3)(a-l)+cs(11-3)2.
(2)
Ii and a are coordinate invariant measures of the deformation and cL(k = 1-5) are material parameters. Specifically, I,=trC and a (=,/{N.C.N}) is the stretch ratio in a muscle fiber direction. C ( = FT. F) is the right Cauchy-Green deformation tensor and N is a unit vector that defines the muscle fiber (i.e. preferred) direction in the undeformed state. For completeness, note that the Cauchy stress t is calculated from W, namely t= -pI+2;;B+;;;F*[email protected], 1
(3)
where p is a Lagrange multiplier that enforces incompressibility and B ( = F. FT) is the left Cauchy-Green deformation tensor. As aforementioned, we assumed that N varied linearly through the thickness of the thin slabs. Hence, we let N=cos0(X3)E,+sincD(X3)E2,
(4)
*Stretch ratios are calculated, namely (ds/d# = A. C. A where ds and dS are infinitesimal lengths in deformed and undeformed states, respectively, and A is a unit vector that defines the direction of interest in the undeformed state. C=FT.F.
specimen and - H/2
(tij)=i
/2 f -h/2
[tijl dX37
(6)
where h(=A,H) is the deformed thickness; Is the out-of-plane stretch ratio was calculated assuming incompressibility. The integration in equation (6) was performed numerically within the regression software using a Romberg method. Best-fit values of the five material parameters C~ were determined separately for each specimen using combined data sets. Data from the loading portion of one cycle of equibiaxial and constant Ai z 1.1,1.15 and 1.2 protocols were combined into a single regression file and used to find a single set of parameter values. Since we had three cycles of data from each of these four protocols, we determined all 34=81 possible sets of best-fit parameters for each specimen. These multiple parameter sets reveal possible variations in material behavior from cycle to cycle (Humphrey et al., 19901~). In contrast to individual material parameters, the strain energy stored in a material provides a measure of the overall stiffness. There are two ways to calculate W. One can integrate experimental second BiolaKirchhoff stresses S with respect to associated Green strains E, or one can directly calculate W from the given values of the material parameters. The second method is easier, and preferred from a practical standpoint; specimen-to-specimen or regional comparisons must be made at identical states of deformation, which are impossible to achieve experimentally. Thus, we used equation (2) and best-fit values of the material parameters to calculate the strain energy that would be stored during equibiaxial and ‘uniaxial’ extensions (x1 =u2=O). Values of W were calculated for equibiaxial stretches of Ii =A2 = 1.05, 1.15, and 1.25 and for uniaxial stretches (1, or A,) of 1.05, 1.15, and 1.25, while the stretch in the orthogonal direction (1, or a,) was fixed at 1.0. Moreover, since the ratio of fiber to cross-fiber energy storage during ‘paired’ uniaxial tests provides an indication of anisotropy, these ratios were compared for uniaxial stretches of 1.05,1.15, and 1.25.
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EOUlelAXlALk-lam
To assess whether there were statistically significant differences in energy storage, we applied a two-way repeated-measures ANOVA to the results with region (inner, middle and outer LV) and protocol (15% equibiaxial stretch and 25% uniaxial stretches) as levels tested, followed by multiple pairwise comparisons using the Duncan q-test. Statistical comparisons were restricted to these protocols since they contained groups of equal variances; groups based on combinations of other protocols had unequal variances since the magnitude of Wdepends on both the protocol and amount of stretch. The energy stored by septal specimens was compared separately to the LV regions using unpaired t-tests.
20
SEPTUU
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A
1-3 j 4;. 1.16
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I 56
20- INNER LV
.
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:*
B
c E m i$ $5
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5 4
RESULTS
0. 1.00
1.18
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Experimental data
We found essentially linear distributions of muscle fibers @(X3), including some that were approximately uniform (i.e. *7”), through the thickness of our 12 histologically examined specimens. This supports our use of equations (4) and (S), and hence calculation of fiber splay as J@T-@e(/H. Mean values of these fiber splays, as well as specimen dimensions, were similar from region to region (Table 1). The Cauchy stress-stretch data, from the loading portion of a single cycle of an equibiaxial stretching test, from all 28 specimens reveal that the mechanical behavior was also qualitatively similar from region to region (Figs 1 and 2; panels A, B, C and D contain data from septal (n =8), and inner (n= S), middle (n =9) and outer (n=6) LV specimens, respectively). These data also show that the behavior was similar in predominant fiber (tll vs Al) and cross-fiber (tzz vs AZ) directions. Constitutive relation
Best-fit values of material parameters [ck in equation (2)] were tabulated for one representative specimen per region (Table 2) and plotted, by region, for all 28 specimens [Fig. 3(A-E)]. Recall that we calculated 2268 (=28 x 81) sets of best-fit ck. Small clusters of parameter values (81 each) within a region reveal cycle-to-cycle variations for one specimen (Fig. 3). The nonoverlap of some of these clusters suggest significant specimen-to-specimen variations in values of the parameters within a region. Conversely, overlap of clouds of parameters between regions suggests that
2o 1 OUTER
LV
0
1.00
1.16 STRETCH
I.56
(A)
Fig. 1. Equibiaxial Cauchy stress-stretch (t,, vs A,) data in the predominant muscle fiber direction. Panels A, B, C and D contain data from the middle portion of the interventricular septum (n = 8), and the inner (n = 5), middle (n = 9) and outer (n = 6) portions of the lateral LV free wall, respectively. For clarity, we show only one cycle of loading data per specimen, and a reduced number of data points. Notice the qualitative similarity in the data from region to region, and that specimen-to-specimen variations were greater in the mid-septum and inner LV. there are no significant variations in these parameters by region; this does not imply, however, that there are no differences in regional properties. Correlation coefficients (RI, R,) between experimental and theoretical Cauchy stresses (Table 2) in-
Table 1. Values (mean& S.D.) of the stress-free outer dimensions L1 and L1, thickness H and fiber splay (in deg mm- I), for all 28 specimens, by region and relative to the undeformed configuration Region Midseptum Inner LV Middle LV Outer LV
& (cm) 1.77*0.09 1.57*0.15 1.70*0.19 1.82kO.13
I
L2@@
1.94*0.09 1.79kO.30 1.96iO.25 1.94*0.16
H(mm)
2.6kO.4 2.5f0.7 2.3 kO.3 2.2 f 0.2
I%-@d/H
24&6 16&5 16&5 24&5
Regional stresses in the heart
EWI8IAXIAL
407
X-FIBER
Cl
c2
C
I I
c3
;
3
y
5
c4 1
20 v,
.oo
1.36
1.16
D
1 OUTER LV
:I(:
C5
1.I0
1.36
I 16
STRETCH (A)
Fig. 2. Equibiaxial Cauchy stress-stretch (t,, vs AZ) data in the predominant cross-fiber direction. Legend same as in Fig. 1.
dicated that the fit to data was similar for all regions.
Fig. 3. Regional best-fit values of the five material parameters (in kPa) for all 28 specimens. Small clusters within a region correspond to the 81 sets of parameters that were determined for each specimen by fitting all possible combined data sets that consisted of single cycles of data from four different protocols. Clearly, the best-fit values of the parameters were similar from region to region.
This ‘goodness of fit’ is illustrated well by plots of the measured and predicted stress-stretch behavior [e.g. Fig. 4(A-D)]. Panels A, B, C, and D show the fit to equibiaxial and constant AI N 1.1, 1.15 and 1.2 protocols that was achieved with a single set of parameters (R, =0.990, R2=0.!84); data are from a representative specimen from the outer portion of the LV free wall. Although difficult to see, the fit to the predominant fiber direction data in panels B-D is similar to that in
the predominant cross-fiber direction. Results were similar for all specimens and all regions. Finally, the predictive capability of equations (2) and (3) was evaluated using data that were not included in the best-fit regression (e.g. the constant AZ= 1.15 protocol). Although predictions (Fig. 4E) were not as good as thefit to data [Fig. 4(A-D)], they
Table 2. Best-fit values of the material parameters C~ (in kPa) in equation (2) for four different specimens that represented well the behavior exhibited in the four regions studied. RI and R2 are values of the correlations between the experimental and theoretical stresses in the ‘predominant’ fiber and cross-fiber directions, respectively Region Midseptum Inner LV Middle LV Outer LV
cl
c2
c3
c*
c5
2.153 0.511 5.293 4.841
15.01 21.58 1.522 3.388
0.003 0.744 0.431 0.771
-1.356 - 7.348 -7.721 -6.055
5.405 1.029 5.607 6.221
Rl
0.994 0.991 0.991 0.990
RZ
0.993 0.992 0.992 0.984
V. P.
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STRETCH
1.25
1 15
0
1.18
1.00
1.36
00
1.16
1.00
1.18
1.00
._
I CONSlANl
1.36
A*-1
1.36
15
El
STRETCH
(A)
Fig. 4. Fit to data for a representative outer-wall LV specimen, by equations (2) and (3). during equibiaxial and constant Ai = 1.1, 1.15 and 1.2 protocols-panels A, B, C and D, respectively. In addition, the predictivecapabilityof equations (2) and (3) is illustrated in panel E wherein data are from a constant AZ= 1.15 protocol, data which were not included in the parameter estimation.
A,=1
25
STRETCH
h2=1.25
RATIO
Fig. 5. Calculated values (mean), by region, of the pseudostrain energy that would be stored in specimens subjected to five different deformations: 5, 15 and 25% equibiaxial extensions (panel A) and 25% uniaxial extensions in the 1 and 2 directions, respectively (panel B). The orthogonal extension was fixed at 1.00 in the uniaxial calculations. Trends include the outer LV tended to be the stiffest, followed by the inner LV, middle septum and finally the middle LV. Statistically, however, the only significant difference was that the outer LV was stiffer than the middle LV (p=O.O6).
middle LV specimens. Other comparisons did not achieve statistical significance. Finally, ratios of mean fiber to cross-fiber energy storage at 5, 15, and 25% uniaxial extensions were greater than unity in all cases (i.e. regions and amounts of stretch) except for the outer LV at 25% extension wherein the ratio was 0.96. Statistical comparisons of these ratios revealed no significant regional differences in anisotropy. DISCUSSION
were reasonable. Again, results specimens and all regions.
were similar
for all
Stored energy Calculated values of W (mean +S.D.) were analyzed by region for various equibiaxial and uniaxial deformations. Two overall trends were found (Fig. 5). First, inner and outer LV specimens were stiffer than middle LV specimens with the outer LV being the stiffest. For example, at an equibiaxial stretch of 1.25, the energy stored in the inner and outer LV regions was 30 and 43% higher, respectively, than that stored in the middle LV. Second, middle septal specimens tended to be stiffer than those from the middle LV; for example, they stored 14% more energy at a 25% equibiaxial stretch. Statistical analyses revealed further that the energy stored by the outer LV was significantly greater (p=O.O6) than that stored by the
Specimen-to-specimen differences in behavior and fiber splay were more pronounced in the midseptal and inner LV regions than in the middle and outer regions of the LV (Table 1, Figs 1 and 2). These observations may reflect a more complex and variable ultrastructure in the septum and subendocardial LV, or perhaps the added difficulty, due to trabeculations, of consistently obtaining septal and inner LV slices at the same transmural depth. In contrast, the ‘middle’ LV specimens (n=9) exhibited a remarkably similar behavior despite being taken from different depths within the wall (Figs 1C and 2C); this suggests that properties in the middle-third of the LV free wall are homogeneous. Moreover, the overall behavior, including hysteresis (not shown), was similar in the predominant fiber and cross-fiber directions in all four regions. This region-to-region similarity in my-
Regional stresses in the heart ocardial behavior can be contrasted with the markedly different behavior exhibited by ventricular epicardium (Humphrey et al., 199Oa, 1992). The majority of the data also suggested that the fiber direction was stiffer than the cross-fiber direction in each of the four regions. However, data were plotted in predominant rather than true fiber and crossfiber directions, thus one must be careful when assessing the degree of anisotropy graphically. Novak (1992) showed that differences between the predominant fiber and cross-fiber stress-stretch behavior decrease as the through-the-thickness fiber splay increases and as the splay becomes more asymmetric with respect to the midplane of the specimen. This finding is intuitively obvious, since tll will equal tzz during an equibiaxial test when the splay varies linearly from 0 to 90” (see the appendix). Thus, local (i.e. true) and global measures of anisotropy can be very different, and the latter often misleading. Although qualitative information is very useful, true insight into material behavior comes only through rigorous quantification via an appropriate constitutive relation; this requires both identification of a specific functional form and determination of best-fit values of the material parameters. The functional forms of most constitutive relations for soft tissues, including myocardium, are often chosen in a somewhat ad hoc fashion. In contrast, consistent with theoretical guidelines, our constitutive relation was inferred directly from experimental data subject only to assumptions of incompressibility and a transverse isotropy that is describable by a W(I,, a). Equations (2) and (3) are fully three-dimensional, contain shearing terms, are not overparameterized, easily admit convergent parameter values, and account for the nonlinear and anisotropic behavior of myocardium over broader classes of tests than previously reported. Moreover, since our W depends on coordinate invariant measures of the finite deformation, it is easy to employ in stress analyses of the heart using any coordinate system (see Humphrey and Yin, 1989a; Humphrey et al., 1990b,c). None the less, the form of W(I,,a) in equation (2) was determined previously from midwall LV data using ‘constant invariant tests’ wherein the value of one invariant (I, or a) was maintained constant while the other was varied, and vice versa. However, constant a-tests cannot be performed on specimens in which the muscle fiber direction varies through the thickness, and thus separate forms of W probably cannot be rigorously determined for regions other than the true midwall LV and perhaps midwall septum; muscle fiber gradients are likely too great in other regions. Although this is a potential limitation of our approach, using separate forms of a W for each region would complicate analyses considerably and should be resorted to only if absolutely necessary. It is reasonable, therefore, to evaluate whether a single W can describe myocardial behavior equally well in different regions.
409
Our results revealed that the fit to data by equations (2) and (3) was similar for each of the four regions studied (Table 2 and Fig. 4). Moreover, the predicted behavior during constant Al tests (i.e. data not included in the parameter estimations) was similar for each region (e.g. Fig. 4E). We conclude, therefore, that a single form of a constitutive relation can describe myocardial behavior well in the four regions studied, thus separate functional forms of a constitutive relation are likely not needed to describe regional behavior. Strictly speaking, stiffness is a measure of the change of stress with respect to strain, and thereby is tensorial. Complete measures of stiffness could have been calculated here since the three-dimensional constitutive relation was known. None the less, for simplicity, we used the scalar Was a measure of myocardial stiffness since stiffer elastic materials store more energy than compliant ones under the same deformation. Based on stored energy, we observed consistent trends in the data: the inner and outer LV regions were stiffer, on average, than the middle portions of the septum and LV free wall, with the outer LV specimens being the stiffest. Moreover, the differences between the outer and middle LV were statistically significant (p = 0.06). Ratios of energy storage during uniaxial extensions revealed further that the fiber direction was generally stiffer than the cross-fiber direction in each region, over the range of deformations considered, and that the degree of anisotropy was similar in all regions except in the inner LV wherein fiber to cross-fiber energy storage tended to be higher. These observations were consistent with the additional finding that W, = (a w/h) was nonnegative at an equibiaxial stretch of 15% for all specimens. Due to biological variability, regional comparisons are ideally performed in the same animal. We were, however, unable to achieve this ideal. Although we always obtained data from the first specimen that we tested, the onset of contracture prevented us from collecting good data from a second specimen (i.e. region) in 14 of 21 hearts. Since previous experiments revealed a stable behavior for over 2 h in single myocardial specimens (Humphrey et al., 1990b), it was surprising that contracture played such a role in this study. We could perform complete tests on two specimens within 2 h of the arrest: 45 min elapsed from the time of arrest to the first preconditioning cycle on the first specimen and testing required only 30 min, 10 to precondition the specimen and verify the stress-free dimensions and 20 to perform six protocols. Moreover, by ‘stringing-up’ the second specimen in a nearby jig, we could transfer it to the device and begin testing within 10 min of the conclusion of the first. It is possible, however, that cyclic stretching may delay the onset of contracture, and thus contracture may have begun in the second specimen while it was merely immersed prior to testing. This is only speculation. A second limitation relates to sites that were not tested. Although the outer LV specimens were easily
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obtained following the excision of 1.5-2.0 mm of epicardium and myocardium, true subepicardial myocardium (i.e. within the 2 mm just below the epicardium) was not tested; the curvature of the LV makes it difficult to obtain consistently large subepicardial specimens with all of the epicardium removed. Since the outer LV layer that we tested was the stiffest, it is likely that true subepicardial myocardium is even stiffer. This speculation must be checked. Finally, all LV specimens were taken from the equatorial region of the lateral free wall. Separate information on myocardial behavior from apical and basal regions and from anterior and posterior locations would be useful too. Investigating all regions was beyond the scope of the present work. Determination of stress-free dimensions was also problematic. In particular, we found that determinations by a single investigator were repeatable to within l-3%, but those by different investigators (VPN and JDH) often differed by 3-6%. Since these measurements were made at the same time, we identified stress-free states that both investigators agreed upon. An important message, however, is that with possible (if unchecked) 3-6% differences within one specimen in a single laboratory, one would expect greater differences from laboratory to laboratory. This complicates interlaboratory comparisons and suggests a need for establishing standards for soft tissue testing. None the less, we now compare our findings with the limited data in the literature. Demer and Yin (1983) performed uniaxial and biaxial tests on thin slabs of LV myocardium excised from 49 canine hearts. Their data suggest that LV myocardial properties may vary from apex to base and from subendocardium to subepicardium. These findings were not conclusive because the specimens may not have been completely passive and the experimental device could not precisely control the tests. Shacklock (1987) reported data from biaxial loading tests on thin specimens taken from the middle and outer portions of the canine LV free wall. The outerwall specimens contained subepicardial myocardium and epicardium and thus, could not be used to infer regional differences in myocardial properties. That the observed behavior differed markedly between these two regions is consistent with our previous findings that epicardial and myocardial behavior are distinctly different (Humphrey et al., 1990a). Przyklenk et al. (1987) performed uniaxial tests on thin strips of LV canine myocardium taken from various transmural sites. Although uniaxial data are not sufficient for quantifying three-dimensional behavior, their data are qualitatively consistent with ours; specimens containing endocardium or epicardium were stiffer than those containing myocardium alone, and subepicardial myocardium was slightly stiffer than midwall myocardium. Their latter result must be interpreted carefully since specimens were excised in the apex-to-base direction, and thus tests
were not consistently performed in a fiber or crossfiber direction. There are, in addition, data that relate indirectly to regional myocardial properties. Buccino et al. (1969) and Smaill and Hunter (1991) report that there is both more intramyocardial collagen and a denser packing of myocytes in the outer half of the LV free wall than in the inner half. Since the mechanical behavior depends on the constituents that comprise a material, these data appear to be consistent with our present findings that the outer LV is the stiffest. One must remember, however, that material behavior depends upon the composition and organization (i.e. architecture) of the constituents, and hence can only be inferred from detailed structural data, information which is not available at present. There are also reports on the inference of regional properties from measurements of LV pressure and uniaxial strain in intact canine hearts (Kent et al., 1978; Lew et al., 1986; Ling et al., 1979). There are several limitations associated with these studies, two of which were recognized by one of the primary groups that employ this method (Lew et al., 1986). First, one needs measures of stress and strain to infer material properties. Regional stresses in the heart depend not only on LV cavity pressure (local or global), but also on regional geometry and boundary conditions. Hence, relating measured cavity pressures to local strains is not sufficient for inferring regional material properties. Second, the stiffness of a material depends upon the complete three-dimensional strain field. Since the heart experiences complex, multidimensional strains in both diastole and systole (Villerreal et al., 1988; Waldman et al., 1985) measurement of uniaxial strains alone is not sufficient for estimating myocardial properties. Guccione et al. (1991) proposed a method for inferring myocardial properties from LV pressure-strain data that alleviates the two major limitations just mentioned. Best-fit values of the material parameters were determined by minimizing differences between predicted and measured two-dimensional epicardial strains, at various cavity pressures, by solving an associated boundary value problem. In this way, multiaxial strain data are used and stress, not pressure, is related to strain. This approach is a significant advancement over similar methods, but is not without limitations. For example, the functional form of the constitutive relation cannot be determined directly from the data. CONCLUSION
There have been many studies on regional variations in coronary blood flow, oxygen consumption, sarcomere lengths, muscle fiber distributions, adrenergic stimulation, muscle activation, and three-dimensional strains (see Glass et al., 1991). It is surprising, therefore, that there has been so little attention dir-
Regional stresses in the heart ected toward quantifying possible regional variations in the biomechanical properties of cardiac tissue. Hence, our findings provide new information on the regional mechanical behavior of normal noncontracting canine myocardium, information that is essential for subsequent studies on regional disease-induced changes in properties. In particular, the present results, together with our previous findings on ventricular epicardium, suggest that the biaxial mechanical properties of the noncontracting canine heart are heterogeneous. The behavior of ventricular epicardium is significantly different from that of myocardium, which in turn appears to vary transmurally within the LV free wall; outer-wall myocardium is stiffer than that in the underlying midwall (p=O.O6), and may be stiffer than inner-wall myocardium as well. Because of the nonlinear behavior of cardiac tissue, even slight differences in material properties may have significant effects on predicted wall stress. Thus, this issue of myocardial heterogeneity merits further, careful attention.
Acknowledgments-We thank Mr John Downs and MS Kristi Pier for their technical expertise. Partial financial support from NIH grants HL 41130 and HL 33621 are gratefully acknowledged. Portions of this work were performed in partial fulfillment of the M.S. degree by VPN at UMBC.
REFERENCES Buccino, R. A., Harris, E., Spann, J. F. and Sonnenblick, E. H. (1969) Response of myocardial connective tissue to development of experimental hypertrophy. Am. J. Physiol. 216, 425-428. Demer, L. L. and Yin, F. C. P. (1983) Passive biaxial mechanical properties of isolated canine myocardium. J. Physiol. (London) 339,615-630. Downs, J., Halperin, H. R., Humphrey, J. D. and Yin, F. C. P. (1990) An improved video based computer tracking system for soft biomaterials testing. IEEE Trans biomed. Engng 31,903-907.
Glass, L., Hunter, P. J. and McCulloch, A. D. (1991) Theory of Heart. Springer, New York. Guccione, J. M., McCulloch, A. D. and Waldman, L. K. (1991) Passive material properties of intact ventricular myocardium determined from a cylindrical model. ASME J. biomech. Engng 113, 42-55. Huisman, R. M., Sipkema, P., Westerhof, N. and Elzinga, G. (1980) Comparison of models to calculate ventricular wall force. Med. biol. Engng Comput. 18, 133-144. Humphrey, J. D., Strumpf, R. K. and Yin, F. C. P. (199Oa) Biaxial mechanical behavior of excised epicardium. Am. J. Physioi. 259, H 101-H108. Humphrey, J. D., Strumpf, R. K. and Yin, F. C. P. (1990b) Determination of a constitutive relation for passive myocardium: I. A new functional form. ASME J. biomech. Engng 112, 333-339. Humphrey, J. D., Strumpf, R. K. and Yin, F. C. P. (199Oc) Determination of a constitutive relation for passive myocardium: II. Parameter estimation. ASME J. biomech. Engng 112, 340-346. Humphrey, J. D., Strumpf, R. K. and Yin, F. C. P. (1992)
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A constitutive theory for biomembranes: application to epicardial mechanics. ASME J. biomech. Engng 114, 461-466.
Humphrey, J. D. and Yin, F. C. P. (1989a) Constitutive relations and finite deformations of passive cardiac tissue. II. Stress analysis in the left ventricle. Circ. Res. 65, 805-817. Humphrey, J. D. and Yin, F. C. P. (1989b) Biomechanical experiments on excised myocardium: theoretical considerations. J. Biomechanics 22, 377-383. Janz, R. F. and Grimm, A. F. (1972) Finite element model for the mechanical behavior of the left ventricle. Circ. Res. 30, 244-252. Janz, R. F. and Grimm, A. F. (1973) Deformation of the diastolic left ventricle. I. Nonlinear elastic effects. Biophys. J. 13,689-704.
Janz R. F. and Waldron, R. J. (1976) Some implications of a constant fiber stress hypothesis in the diastolic left ventricle. Bull. Math. Biol. 38, 401-413. Kent, R. S., Carew, T. E., LeWinter, M. M. and Covell, J. W. (1978) Comparison of left ventricular free wall and septal diastolic compliance in the dog. Am. J. Physiol. 234, H392-H398. Lew, W. Y. W. and LeWinter, M. M. (1986) Regional comparison of midwall segment and area shortening in the canine left ventricle. C&c. Res. 58, 678-691. Line. D.. Rankin. J. S.. Edwards. C. H.. McHale. P. A. and Anderson, R. W. (1979) Regional diastolic mechanics of the left ventricle in the conscious dog. Am. J. Physiol. 236, H323-H330. Mirsky, I. (1970) Effects of anisotropy and nonhomogeneity on left ventricular stresses in the intact heart. Bull. Math. Biophys. 32, 197-210. Mirsky, I. (1979) Elastic properties of the myocardium: a quantitative approach with physiologic and clinical applications. In Handbook ofPhysiology (Edited by Berne, R. M., Sperelakis, N. and Geigen, S. R.), pp. 497-531. American Physiologic Society, Bethesda, MD. Moriarty, T. F. (1980) The law of Laplace: its limitations as a relation for diastolic pressure, volume or wall stress of the left ventricle. Circ. Res. 46, 321-331. Novak, V. P. (1992) Regional mechanical properties of passive myocardium. M.S. thesis, University of Maryland. Baltimore. Nielsen, P. M. F., LeGrice, I. J., Smaill, B. H. and Hunter, P. J. (1991) Mathematical model of geometry and fibrous structure of the heart. Am. J. Physiol. 260, H1365-H1378. Omens. J. H. and Fung, Y. C. (1990) Residual strain in rat left ventricle. Circ. Res. 66, 37-45. Przyklenk, K., Connelly, C. M., McLaughlin, R. J., Kloner. R. A. and Apstein, C. S. (1987) Effect of myocyte necrosis on strength, strain and stiffness of isolated myocardial strips. Am. Heart J. 114, 134991359. Shacklock, A. (1987) Biaxial testing of cardiac tissue (1987) M.S. thesis, University of Auckland, New Zealand. Smaill, B. H. and Hunter, P. J. (1991) Structure and function of the diastolic heart. In Theory ofHeart (Edited by Glass, L., Hunter, P. J. and McCulloch, A. D.). Springer, New York. Villarreal, F. J., Waldman, L. K. and Lew, W. Y. W. (1988) Technique for measuring regional two-dimensional finite strains in canine left ventricle. Circ. Res. 62, 711-721. Waidman, L. K., Fung, Y. C. and Covell, J. W. (1985) Transmural myocardial deformation in the canine left ventricle: normal in uioo three-dimensional strains. Circ. Res. 57, 152-163. Yin, F. C. P. (1981) Ventricular wall stress. Circ. Res. 49, 829-842. Yin, F. C. P., Strumpf, R. K., Chew, P. H. and Zeger, S. L. (1987) Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading. J. Biomechnnics 20, 577-599.
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APPENDIX For simplicity, consider an equibiaxial test without shear. That is, let ,& =& =1 and K~= K*=O. Thus, even with a linear splay of muscle fibers through the thickness [see equations (4) and (S)] u (=A) and 1, (= 21’ + l/12’) are independent of @, thereby allowing equation (6) to be integrated easily, namely
+w,L
1’
(Al)
I
644
[
1 sin 2mT-sin 2Q ;+ 4(%-W
[
1 sin 2DT-sin 2mB 24(%-W
+w,n
Now, consider the following special cases. Case I: In the limit as C++O and @a-O, equations (Al) and (A2) become (fl,)=fll=2W~(~2-l/124)+W,rl, (t2*)=tZZ=2W~(r22-l/14),
(A3)
which, as we expected, are the same as those in Humphrey et al. (1990b) for the case of no fiber splay and N=(l, 0,O). Similarly, when QT-+ n/2 and @a-+ n/2, one obtains the same results except that t,, is the stress in the fiber direction and t, 1 the stress in the cross-fiber direction. Case II: As 4 -+ n/2 and (Pa+ 0, equations (Al) and (A2) become (tll)=(t22)=2W~(~Z-l/124)+t~IZ,
(A4)
which is again intuitively obvious. That is, biaxial stresses should be the same in an equibiaxial test when the muscle fiber splay changes linearly from 0 to 7r/2. Case Ill: As mT+ j? and @a+ - /?,equations (Al) and (A2) yield
W Thus, as fi increases from 0 to n/2. the difference between (t,,) and (tz2) becomes less and less. That is, differences between the predominant fiber and cross-fiber stresses (e.g. Figs 1 and 2) become less and less with increased symmetric fiber splays even though the true fiber direction is stiffer. This important observation must be remembered when interpreting anisotropy from plots of experimental data.
Pergamon
1994
Copyright 0 1994 Ekvier Science Ltd Printed in Great Britain. All rights reserved OoZl -9290/94 $6.00 + .@I
REGIONAL
MECHANICAL PROPERTIES MYOCARDIUM
OF PASSIVE
V. P. NOVAK, F. C. P. YIN* and J. D. HUMPHREYP Department of Mechanical Engineering, The University of Maryland, Baltimore, MD 21228, U.S.A. and *Departments of Medicine and Biomedical Engineering, The Johns Hopkins Medical Institutions, Baltimore, MD 21205, U.S.A. Abstrati-There is considerable interest in calculating regional stresses in the heart. This, in turn, requires complete information on regional material properties which, surprisingly, is not available. The specific aim of this work, therefore, was to determine if transmural differences exist in the mechanical behavior of passive myocardium in the equatorial region of the heart. Thus, we performed in uitro biaxial experiments on 28 thin, rectangular slabs of noncontracting myocardium excised from four regions within canine hearts: the middle portion of the interventricular septum (n = 8), and the inner (n = 5), middle (n = 9) and outer (n = 6) layers of the lateral left ventricular (LV) free wall. There were three major findings. First, an existing threedimensional constitutive relation described the nonlinear and anisotropic behavior exhibited in the four regions equally well. Second, the anisotropy was similar in each region. Third, there were, however, regional differences in the strain-energy stored by specimens during identical finite deformations. In particular, specimens from inner and outer portions of the LV free wall tended to be stiffer than those from the middle of the LV free wall and septum. These findings, together with previous results on excised eoicardium. suggest that the mechanical uroDerties of the heart are qualitatively similar from region to rkgion, but quantitatively different.
INTRODUCTION A detailed understanding of cardiac physiology and pathophysiology requires quantification of regional phenomena, including distributions of stress and strain. Biplane cineradiography, one- and two-dimensional sonomicrometry, and most recently magnetic resonance imaging (MRI) provide important information on regional strains in the intact beating heart. In contrast, regional stresses cannot be measured reliably with the current methods. Rather, they must be calculated using the methods of continuum mechanics, which in turn requires knowledge of regional mechanical properties. The need for data on the regional biomechanical behavior of cardiac tissue has been known for many years, as has the potential importance of this information. For example, Mirsky (1970) showed that predicted transmural wall stresses differ significantly when myocardium is assumed to be heterogeneous rather than homogeneous as assumed by most. Similar findings were obtained by Janz and Grimm (1972, 1973) using element analyses of the left ventricle. Janz and Waldron (1976) and Moriarty (1980) later hypothesized that circumferential wall stress should be nearly uniform across the wall of the heart at end diastole, not much larger at the endocardium as predicted by most early analyses (Huisman et al., 1980). Using simplified models, these investigators showed that a uniform transmural stress distribution could exist in the heart if values of a circumferential modulus increased monotonically from the endocar-
dium to the epicardium. Although their specific results must be viewed cautiously-they neglected anisotropy due to transmural fiber distributions (Humphrey and Yin, 1989a), residual stress (Omens and Fung, 1990) and twist in the passive heart (Guccione et al., 1991), each of which significantly affects predicted transmural stress distributions-implications of heterogeneous wall properties were clearly demonstrated. Despite these provocative findings as well as direct calls for data (Huisman et al., 1980; Mirsky, 1979; Yin, 1981), there remains little information on the regional mechanical properties of cardiac tissue. The goal of this work, therefore, was to determine if transmural differences exist in the mechanical behavior of passive myocardium in the equatorial region of the canine heart. Consequently, (a) we performed in oitro biaxial experiments on noncontracting myocardium excised from four regions within canine hearts: the middle of the interventricular septum and inner, middle and outer portions of the lateral left ventricular (LV) free wall; (b) we quantified the observed biomechanical behavior using an existing three-dimensional nonlinear, anisotropic constitutive relation; (c) we compared the pseudostrain-energy stored, at identical deformations, by specimens from each of the four regions. This study provides, therefore, the first detailed information on the regional multiaxial stress-strain behavior of noncontracting myocardium. METHODS Specimen preparation
Received in final form 20 August 1993. t Author to whom correspondence should be addressed.
Mongrel dogs (u 20 kg) were anesthetized with sodium pentobarbital(30 mg kg- ‘), intubated and ven403
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The LV free wall and interventricular septum were then excised from the isolated heart, and the papillary muscles removed. Thin rectangular slabs ( N 2 cm x 2 cm x 0.25 cm) of myocardium were cut sequentially from both the lateral LV free wall and the septum, tangential to the epicardial or endocardial surfaces, using a rotary meat slicer. Although we obtained and tested multiple slabs from each heart, the onset of contracture prevented us from collecting ‘good’ data from a second specimen (i.e. region) in 14 of 21 hearts. Thus, we report data from 28 specimens taken from 21 hearts: 8 from the middle of the interventricular septum and 5, 9 and 6 specimens from inner, middle and outer portions of the lateral LV free wall, respectively. ‘Middle’ means within the middle-third of a wall, ‘inner LV’ corresponds to the subendocardium, just below the deepest invagination in the endocardial surface, and ‘outer LV specimens were taken N 1.5-2.0 mm below the epicardial surface.
to place specimens in the device such that the fiber splay (as judged by the eye) was symmetric with regard to the midplane; this reduces through-the-thickness variations in stresses (Humphrey and Yin, 1989b). Each specimen was coupled to the four carriages of the device using four continuous strands of 000 suture, each strand looped five times through one side of the specimen. Uniform spacing between each loop was ensured by sewing through a paper template that was placed atop the specimen; this template was removed prior to testing. A central region of the specimen was darkened with india ink, and four 1 mm diameter steel balls (4 mg each) were glued onto the specimen in a rectangular pattern within the central region; these balls demarcated -3% of the planar surface area. A white light source, directed at an angle to the surface of the specimen, produced a small bright spot on each ball that was tracked with custom video-based software (Downs et al., 1990). Once mounted, specimens were subjected to an initial preconditioning protocol that consisted of 7-10 cycles of equibiaxial loading (lo-45 g) at 0.1 Hz. These preconditioning protocols minimized nonrepeatable, history-dependent effects that result from excising, slicing and mounting a specimen. The unloaded, stress-free state of each specimen (i.e. outer dimensions and locations of the four tracking markers) was recorded following this preconditioning, and was checked three times to ensure repeatability. Each specimen was then subjected to five different cyclic stretching protocols conducted at 0.1 Hz: equibiaxial stretching (5-27%) and tests wherein the stretch in one direction was maintained constant at a prescribed value (‘v 10, 15 or 20%), while the orthogonal direction was cyclically stretched (5-27%). The equibiaxial protocol was performed first, and was repeated after the other protocols were finished. The first 5-7 cycles preconditioned the tissue for each specific protocol, whereas data were collected at 30 Hz from the last three cycles of each protocol. Specimen thickness was measured in the central region, at the completion of testing, using dial calipers. Specimens were then fixed in the unloaded state in a 10% buffered formalin, and stored at 5°C.
Biaxial tests
Fiber angles
Our test system and experimental methods are described elsewhere (Humphrey et al., 1990a, b; Yin et al., 1987). Briefly, a 386 PC controls two pairs of orthogonally mounted carriages via two independent sets of linear motors, worm gears and lead screws. A force transducer, riding on one carriage of each pair, measures the in-plane applied loads while a video camera, mounted above the specimen, monitors the deformation within a central region of the specimen. Resolution was 3 mN and 0.004 mm, respectively. Each rectangular specimen was immersed in the oxygenated cardioplegic solution at room temperature within the test chamber. In particular, we tried
Detailed information on orientations of muscle fibers in the canine heart is in Nielsen et al. (1991). Their data strongly suggest that for the regions studied herein, we could assume that the fiber splay was linear through the thickness of our thin (2-3 mm) slabs. Thus, we measured the predominant fiber angle on the top and bottom surfaces (QT and @a) in the central region of each fixed specimen. DT and @a were measured with a 10 x dissecting microscope and interactive software by recording coordinates along muscle fibers relative to orthogonal fiducial marks that were introduced in the specimen using a jig containing three heated 20 g needles. In addition, we measured through-the-thickness fiber distributions in
tilated. The chest was entered via a midline thoracotomy, and the pericardium opened. A 7F pigtail catheter was advanced from the right common carotid artery to just above the aortic valve. The subclavian and brachiocephalic arteries were ligated, the descending aorta cross-clamped, and the right atrium and left ventricle vented to atmosphere. The heart was then arrested in diastole with infusions of a cold heparinized (8000 units) oxygenated cardioplegic solution (in mM): 0.4 KH,PO*, 1.0 MgS04. 7H20, 28 NaHCO,, 1.5 CaClz. 2H,O, 5.6 dextrose, 117.5 L-glutamic acid and 1.0 adenosine which was used to dilate the coronary bed. Although 30 ml of solution typically induced cardiac arrest, 240-300 ml were infused to ensure thorough myocardial perfusion; return flow of solution was always evident in the epicardial coronary veins. The heart was then rapidly excised, and the circumflex, left anterior descending and septal coronary arteries perfused further with cardioplegic solution (N 120 ml). This resulted in an almost complete washout of blood from the myocardium. All animal care and use was IACUC approved, and thus consistent with the NIH Guide for the Care and Use of Laboratory Animals.
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Regional stresses in the heart
sequential 100 pm hematoxylin- and eosin-stained sections in 12 (three randomly selected from each region) of the 28 specimens, again using light microscopy. Data analysis
where El and Ez are orthonormal ing directions, and
bases in the stretch-
(5) H is the thickness of the undeformed
It is straightforward to calculate experimental biaxial stresses and strains from stress-free specimen dimensions, the two in-plane applied loads and the motions of four tracking markers (Humphrey et al., 1990a). Briefly, ‘measurable’ quantities of interest are the in-plane normal components (tll, tzz) of the Cauchy stress t and the in-plane components (&, rci, &, rc2) of the deformation gradient F. Subscripts 1 and 2 denote in-plane biaxial stretching directions (i.e. predominant fiber and cross-fiber directions, respectively). In the figures below, however, we plot Cauchy stresses vs the stretch ratios in the 1 and 2 directions:* Ai =J_,
A2 = dm.
(1)
To quantify the behavior of any material requires identification of a constitutive relation. For materials like passive myocardium, which exhibit complex nonlinear anisotropic behavior under finite strains, constitutive relations are often best posed in terms of a pseudostrain energy W. That is, a scalar function that represents the energy stored in a material during deformation. We used a W that we previously identified from biaxial tests on noncontracting canine myocardium excised from the midwall of the LV free wall (Humphrey et al., 1990b):
+c.+(11-3)(a-l)+cs(11-3)2.
(2)
Ii and a are coordinate invariant measures of the deformation and cL(k = 1-5) are material parameters. Specifically, I,=trC and a (=,/{N.C.N}) is the stretch ratio in a muscle fiber direction. C ( = FT. F) is the right Cauchy-Green deformation tensor and N is a unit vector that defines the muscle fiber (i.e. preferred) direction in the undeformed state. For completeness, note that the Cauchy stress t is calculated from W, namely t= -pI+2;;B+;;;F*[email protected], 1
(3)
where p is a Lagrange multiplier that enforces incompressibility and B ( = F. FT) is the left Cauchy-Green deformation tensor. As aforementioned, we assumed that N varied linearly through the thickness of the thin slabs. Hence, we let N=cos0(X3)E,+sincD(X3)E2,
(4)
*Stretch ratios are calculated, namely (ds/d# = A. C. A where ds and dS are infinitesimal lengths in deformed and undeformed states, respectively, and A is a unit vector that defines the direction of interest in the undeformed state. C=FT.F.
specimen and - H/2
(tij)=i
/2 f -h/2
[tijl dX37
(6)
where h(=A,H) is the deformed thickness; Is the out-of-plane stretch ratio was calculated assuming incompressibility. The integration in equation (6) was performed numerically within the regression software using a Romberg method. Best-fit values of the five material parameters C~ were determined separately for each specimen using combined data sets. Data from the loading portion of one cycle of equibiaxial and constant Ai z 1.1,1.15 and 1.2 protocols were combined into a single regression file and used to find a single set of parameter values. Since we had three cycles of data from each of these four protocols, we determined all 34=81 possible sets of best-fit parameters for each specimen. These multiple parameter sets reveal possible variations in material behavior from cycle to cycle (Humphrey et al., 19901~). In contrast to individual material parameters, the strain energy stored in a material provides a measure of the overall stiffness. There are two ways to calculate W. One can integrate experimental second BiolaKirchhoff stresses S with respect to associated Green strains E, or one can directly calculate W from the given values of the material parameters. The second method is easier, and preferred from a practical standpoint; specimen-to-specimen or regional comparisons must be made at identical states of deformation, which are impossible to achieve experimentally. Thus, we used equation (2) and best-fit values of the material parameters to calculate the strain energy that would be stored during equibiaxial and ‘uniaxial’ extensions (x1 =u2=O). Values of W were calculated for equibiaxial stretches of Ii =A2 = 1.05, 1.15, and 1.25 and for uniaxial stretches (1, or A,) of 1.05, 1.15, and 1.25, while the stretch in the orthogonal direction (1, or a,) was fixed at 1.0. Moreover, since the ratio of fiber to cross-fiber energy storage during ‘paired’ uniaxial tests provides an indication of anisotropy, these ratios were compared for uniaxial stretches of 1.05,1.15, and 1.25.
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et al.
EOUlelAXlALk-lam
To assess whether there were statistically significant differences in energy storage, we applied a two-way repeated-measures ANOVA to the results with region (inner, middle and outer LV) and protocol (15% equibiaxial stretch and 25% uniaxial stretches) as levels tested, followed by multiple pairwise comparisons using the Duncan q-test. Statistical comparisons were restricted to these protocols since they contained groups of equal variances; groups based on combinations of other protocols had unequal variances since the magnitude of Wdepends on both the protocol and amount of stretch. The energy stored by septal specimens was compared separately to the LV regions using unpaired t-tests.
20
SEPTUU
ln
.
A
1-3 j 4;. 1.16
1.00
I 56
20- INNER LV
.
v)
:*
B
c E m i$ $5
10..
5 4
RESULTS
0. 1.00
1.18
1
i.00
l.IS
1.b6
Experimental data
We found essentially linear distributions of muscle fibers @(X3), including some that were approximately uniform (i.e. *7”), through the thickness of our 12 histologically examined specimens. This supports our use of equations (4) and (S), and hence calculation of fiber splay as J@T-@e(/H. Mean values of these fiber splays, as well as specimen dimensions, were similar from region to region (Table 1). The Cauchy stress-stretch data, from the loading portion of a single cycle of an equibiaxial stretching test, from all 28 specimens reveal that the mechanical behavior was also qualitatively similar from region to region (Figs 1 and 2; panels A, B, C and D contain data from septal (n =8), and inner (n= S), middle (n =9) and outer (n=6) LV specimens, respectively). These data also show that the behavior was similar in predominant fiber (tll vs Al) and cross-fiber (tzz vs AZ) directions. Constitutive relation
Best-fit values of material parameters [ck in equation (2)] were tabulated for one representative specimen per region (Table 2) and plotted, by region, for all 28 specimens [Fig. 3(A-E)]. Recall that we calculated 2268 (=28 x 81) sets of best-fit ck. Small clusters of parameter values (81 each) within a region reveal cycle-to-cycle variations for one specimen (Fig. 3). The nonoverlap of some of these clusters suggest significant specimen-to-specimen variations in values of the parameters within a region. Conversely, overlap of clouds of parameters between regions suggests that
2o 1 OUTER
LV
0
1.00
1.16 STRETCH
I.56
(A)
Fig. 1. Equibiaxial Cauchy stress-stretch (t,, vs A,) data in the predominant muscle fiber direction. Panels A, B, C and D contain data from the middle portion of the interventricular septum (n = 8), and the inner (n = 5), middle (n = 9) and outer (n = 6) portions of the lateral LV free wall, respectively. For clarity, we show only one cycle of loading data per specimen, and a reduced number of data points. Notice the qualitative similarity in the data from region to region, and that specimen-to-specimen variations were greater in the mid-septum and inner LV. there are no significant variations in these parameters by region; this does not imply, however, that there are no differences in regional properties. Correlation coefficients (RI, R,) between experimental and theoretical Cauchy stresses (Table 2) in-
Table 1. Values (mean& S.D.) of the stress-free outer dimensions L1 and L1, thickness H and fiber splay (in deg mm- I), for all 28 specimens, by region and relative to the undeformed configuration Region Midseptum Inner LV Middle LV Outer LV
& (cm) 1.77*0.09 1.57*0.15 1.70*0.19 1.82kO.13
I
L2@@
1.94*0.09 1.79kO.30 1.96iO.25 1.94*0.16
H(mm)
2.6kO.4 2.5f0.7 2.3 kO.3 2.2 f 0.2
I%-@d/H
24&6 16&5 16&5 24&5
Regional stresses in the heart
EWI8IAXIAL
407
X-FIBER
Cl
c2
C
I I
c3
;
3
y
5
c4 1
20 v,
.oo
1.36
1.16
D
1 OUTER LV
:I(:
C5
1.I0
1.36
I 16
STRETCH (A)
Fig. 2. Equibiaxial Cauchy stress-stretch (t,, vs AZ) data in the predominant cross-fiber direction. Legend same as in Fig. 1.
dicated that the fit to data was similar for all regions.
Fig. 3. Regional best-fit values of the five material parameters (in kPa) for all 28 specimens. Small clusters within a region correspond to the 81 sets of parameters that were determined for each specimen by fitting all possible combined data sets that consisted of single cycles of data from four different protocols. Clearly, the best-fit values of the parameters were similar from region to region.
This ‘goodness of fit’ is illustrated well by plots of the measured and predicted stress-stretch behavior [e.g. Fig. 4(A-D)]. Panels A, B, C, and D show the fit to equibiaxial and constant AI N 1.1, 1.15 and 1.2 protocols that was achieved with a single set of parameters (R, =0.990, R2=0.!84); data are from a representative specimen from the outer portion of the LV free wall. Although difficult to see, the fit to the predominant fiber direction data in panels B-D is similar to that in
the predominant cross-fiber direction. Results were similar for all specimens and all regions. Finally, the predictive capability of equations (2) and (3) was evaluated using data that were not included in the best-fit regression (e.g. the constant AZ= 1.15 protocol). Although predictions (Fig. 4E) were not as good as thefit to data [Fig. 4(A-D)], they
Table 2. Best-fit values of the material parameters C~ (in kPa) in equation (2) for four different specimens that represented well the behavior exhibited in the four regions studied. RI and R2 are values of the correlations between the experimental and theoretical stresses in the ‘predominant’ fiber and cross-fiber directions, respectively Region Midseptum Inner LV Middle LV Outer LV
cl
c2
c3
c*
c5
2.153 0.511 5.293 4.841
15.01 21.58 1.522 3.388
0.003 0.744 0.431 0.771
-1.356 - 7.348 -7.721 -6.055
5.405 1.029 5.607 6.221
Rl
0.994 0.991 0.991 0.990
RZ
0.993 0.992 0.992 0.984
V. P.
408
NOVAK
et al.
STRETCH
1.25
1 15
0
1.18
1.00
1.36
00
1.16
1.00
1.18
1.00
._
I CONSlANl
1.36
A*-1
1.36
15
El
STRETCH
(A)
Fig. 4. Fit to data for a representative outer-wall LV specimen, by equations (2) and (3). during equibiaxial and constant Ai = 1.1, 1.15 and 1.2 protocols-panels A, B, C and D, respectively. In addition, the predictivecapabilityof equations (2) and (3) is illustrated in panel E wherein data are from a constant AZ= 1.15 protocol, data which were not included in the parameter estimation.
A,=1
25
STRETCH
h2=1.25
RATIO
Fig. 5. Calculated values (mean), by region, of the pseudostrain energy that would be stored in specimens subjected to five different deformations: 5, 15 and 25% equibiaxial extensions (panel A) and 25% uniaxial extensions in the 1 and 2 directions, respectively (panel B). The orthogonal extension was fixed at 1.00 in the uniaxial calculations. Trends include the outer LV tended to be the stiffest, followed by the inner LV, middle septum and finally the middle LV. Statistically, however, the only significant difference was that the outer LV was stiffer than the middle LV (p=O.O6).
middle LV specimens. Other comparisons did not achieve statistical significance. Finally, ratios of mean fiber to cross-fiber energy storage at 5, 15, and 25% uniaxial extensions were greater than unity in all cases (i.e. regions and amounts of stretch) except for the outer LV at 25% extension wherein the ratio was 0.96. Statistical comparisons of these ratios revealed no significant regional differences in anisotropy. DISCUSSION
were reasonable. Again, results specimens and all regions.
were similar
for all
Stored energy Calculated values of W (mean +S.D.) were analyzed by region for various equibiaxial and uniaxial deformations. Two overall trends were found (Fig. 5). First, inner and outer LV specimens were stiffer than middle LV specimens with the outer LV being the stiffest. For example, at an equibiaxial stretch of 1.25, the energy stored in the inner and outer LV regions was 30 and 43% higher, respectively, than that stored in the middle LV. Second, middle septal specimens tended to be stiffer than those from the middle LV; for example, they stored 14% more energy at a 25% equibiaxial stretch. Statistical analyses revealed further that the energy stored by the outer LV was significantly greater (p=O.O6) than that stored by the
Specimen-to-specimen differences in behavior and fiber splay were more pronounced in the midseptal and inner LV regions than in the middle and outer regions of the LV (Table 1, Figs 1 and 2). These observations may reflect a more complex and variable ultrastructure in the septum and subendocardial LV, or perhaps the added difficulty, due to trabeculations, of consistently obtaining septal and inner LV slices at the same transmural depth. In contrast, the ‘middle’ LV specimens (n=9) exhibited a remarkably similar behavior despite being taken from different depths within the wall (Figs 1C and 2C); this suggests that properties in the middle-third of the LV free wall are homogeneous. Moreover, the overall behavior, including hysteresis (not shown), was similar in the predominant fiber and cross-fiber directions in all four regions. This region-to-region similarity in my-
Regional stresses in the heart ocardial behavior can be contrasted with the markedly different behavior exhibited by ventricular epicardium (Humphrey et al., 199Oa, 1992). The majority of the data also suggested that the fiber direction was stiffer than the cross-fiber direction in each of the four regions. However, data were plotted in predominant rather than true fiber and crossfiber directions, thus one must be careful when assessing the degree of anisotropy graphically. Novak (1992) showed that differences between the predominant fiber and cross-fiber stress-stretch behavior decrease as the through-the-thickness fiber splay increases and as the splay becomes more asymmetric with respect to the midplane of the specimen. This finding is intuitively obvious, since tll will equal tzz during an equibiaxial test when the splay varies linearly from 0 to 90” (see the appendix). Thus, local (i.e. true) and global measures of anisotropy can be very different, and the latter often misleading. Although qualitative information is very useful, true insight into material behavior comes only through rigorous quantification via an appropriate constitutive relation; this requires both identification of a specific functional form and determination of best-fit values of the material parameters. The functional forms of most constitutive relations for soft tissues, including myocardium, are often chosen in a somewhat ad hoc fashion. In contrast, consistent with theoretical guidelines, our constitutive relation was inferred directly from experimental data subject only to assumptions of incompressibility and a transverse isotropy that is describable by a W(I,, a). Equations (2) and (3) are fully three-dimensional, contain shearing terms, are not overparameterized, easily admit convergent parameter values, and account for the nonlinear and anisotropic behavior of myocardium over broader classes of tests than previously reported. Moreover, since our W depends on coordinate invariant measures of the finite deformation, it is easy to employ in stress analyses of the heart using any coordinate system (see Humphrey and Yin, 1989a; Humphrey et al., 1990b,c). None the less, the form of W(I,,a) in equation (2) was determined previously from midwall LV data using ‘constant invariant tests’ wherein the value of one invariant (I, or a) was maintained constant while the other was varied, and vice versa. However, constant a-tests cannot be performed on specimens in which the muscle fiber direction varies through the thickness, and thus separate forms of W probably cannot be rigorously determined for regions other than the true midwall LV and perhaps midwall septum; muscle fiber gradients are likely too great in other regions. Although this is a potential limitation of our approach, using separate forms of a W for each region would complicate analyses considerably and should be resorted to only if absolutely necessary. It is reasonable, therefore, to evaluate whether a single W can describe myocardial behavior equally well in different regions.
409
Our results revealed that the fit to data by equations (2) and (3) was similar for each of the four regions studied (Table 2 and Fig. 4). Moreover, the predicted behavior during constant Al tests (i.e. data not included in the parameter estimations) was similar for each region (e.g. Fig. 4E). We conclude, therefore, that a single form of a constitutive relation can describe myocardial behavior well in the four regions studied, thus separate functional forms of a constitutive relation are likely not needed to describe regional behavior. Strictly speaking, stiffness is a measure of the change of stress with respect to strain, and thereby is tensorial. Complete measures of stiffness could have been calculated here since the three-dimensional constitutive relation was known. None the less, for simplicity, we used the scalar Was a measure of myocardial stiffness since stiffer elastic materials store more energy than compliant ones under the same deformation. Based on stored energy, we observed consistent trends in the data: the inner and outer LV regions were stiffer, on average, than the middle portions of the septum and LV free wall, with the outer LV specimens being the stiffest. Moreover, the differences between the outer and middle LV were statistically significant (p = 0.06). Ratios of energy storage during uniaxial extensions revealed further that the fiber direction was generally stiffer than the cross-fiber direction in each region, over the range of deformations considered, and that the degree of anisotropy was similar in all regions except in the inner LV wherein fiber to cross-fiber energy storage tended to be higher. These observations were consistent with the additional finding that W, = (a w/h) was nonnegative at an equibiaxial stretch of 15% for all specimens. Due to biological variability, regional comparisons are ideally performed in the same animal. We were, however, unable to achieve this ideal. Although we always obtained data from the first specimen that we tested, the onset of contracture prevented us from collecting good data from a second specimen (i.e. region) in 14 of 21 hearts. Since previous experiments revealed a stable behavior for over 2 h in single myocardial specimens (Humphrey et al., 1990b), it was surprising that contracture played such a role in this study. We could perform complete tests on two specimens within 2 h of the arrest: 45 min elapsed from the time of arrest to the first preconditioning cycle on the first specimen and testing required only 30 min, 10 to precondition the specimen and verify the stress-free dimensions and 20 to perform six protocols. Moreover, by ‘stringing-up’ the second specimen in a nearby jig, we could transfer it to the device and begin testing within 10 min of the conclusion of the first. It is possible, however, that cyclic stretching may delay the onset of contracture, and thus contracture may have begun in the second specimen while it was merely immersed prior to testing. This is only speculation. A second limitation relates to sites that were not tested. Although the outer LV specimens were easily
410
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obtained following the excision of 1.5-2.0 mm of epicardium and myocardium, true subepicardial myocardium (i.e. within the 2 mm just below the epicardium) was not tested; the curvature of the LV makes it difficult to obtain consistently large subepicardial specimens with all of the epicardium removed. Since the outer LV layer that we tested was the stiffest, it is likely that true subepicardial myocardium is even stiffer. This speculation must be checked. Finally, all LV specimens were taken from the equatorial region of the lateral free wall. Separate information on myocardial behavior from apical and basal regions and from anterior and posterior locations would be useful too. Investigating all regions was beyond the scope of the present work. Determination of stress-free dimensions was also problematic. In particular, we found that determinations by a single investigator were repeatable to within l-3%, but those by different investigators (VPN and JDH) often differed by 3-6%. Since these measurements were made at the same time, we identified stress-free states that both investigators agreed upon. An important message, however, is that with possible (if unchecked) 3-6% differences within one specimen in a single laboratory, one would expect greater differences from laboratory to laboratory. This complicates interlaboratory comparisons and suggests a need for establishing standards for soft tissue testing. None the less, we now compare our findings with the limited data in the literature. Demer and Yin (1983) performed uniaxial and biaxial tests on thin slabs of LV myocardium excised from 49 canine hearts. Their data suggest that LV myocardial properties may vary from apex to base and from subendocardium to subepicardium. These findings were not conclusive because the specimens may not have been completely passive and the experimental device could not precisely control the tests. Shacklock (1987) reported data from biaxial loading tests on thin specimens taken from the middle and outer portions of the canine LV free wall. The outerwall specimens contained subepicardial myocardium and epicardium and thus, could not be used to infer regional differences in myocardial properties. That the observed behavior differed markedly between these two regions is consistent with our previous findings that epicardial and myocardial behavior are distinctly different (Humphrey et al., 1990a). Przyklenk et al. (1987) performed uniaxial tests on thin strips of LV canine myocardium taken from various transmural sites. Although uniaxial data are not sufficient for quantifying three-dimensional behavior, their data are qualitatively consistent with ours; specimens containing endocardium or epicardium were stiffer than those containing myocardium alone, and subepicardial myocardium was slightly stiffer than midwall myocardium. Their latter result must be interpreted carefully since specimens were excised in the apex-to-base direction, and thus tests
were not consistently performed in a fiber or crossfiber direction. There are, in addition, data that relate indirectly to regional myocardial properties. Buccino et al. (1969) and Smaill and Hunter (1991) report that there is both more intramyocardial collagen and a denser packing of myocytes in the outer half of the LV free wall than in the inner half. Since the mechanical behavior depends on the constituents that comprise a material, these data appear to be consistent with our present findings that the outer LV is the stiffest. One must remember, however, that material behavior depends upon the composition and organization (i.e. architecture) of the constituents, and hence can only be inferred from detailed structural data, information which is not available at present. There are also reports on the inference of regional properties from measurements of LV pressure and uniaxial strain in intact canine hearts (Kent et al., 1978; Lew et al., 1986; Ling et al., 1979). There are several limitations associated with these studies, two of which were recognized by one of the primary groups that employ this method (Lew et al., 1986). First, one needs measures of stress and strain to infer material properties. Regional stresses in the heart depend not only on LV cavity pressure (local or global), but also on regional geometry and boundary conditions. Hence, relating measured cavity pressures to local strains is not sufficient for inferring regional material properties. Second, the stiffness of a material depends upon the complete three-dimensional strain field. Since the heart experiences complex, multidimensional strains in both diastole and systole (Villerreal et al., 1988; Waldman et al., 1985) measurement of uniaxial strains alone is not sufficient for estimating myocardial properties. Guccione et al. (1991) proposed a method for inferring myocardial properties from LV pressure-strain data that alleviates the two major limitations just mentioned. Best-fit values of the material parameters were determined by minimizing differences between predicted and measured two-dimensional epicardial strains, at various cavity pressures, by solving an associated boundary value problem. In this way, multiaxial strain data are used and stress, not pressure, is related to strain. This approach is a significant advancement over similar methods, but is not without limitations. For example, the functional form of the constitutive relation cannot be determined directly from the data. CONCLUSION
There have been many studies on regional variations in coronary blood flow, oxygen consumption, sarcomere lengths, muscle fiber distributions, adrenergic stimulation, muscle activation, and three-dimensional strains (see Glass et al., 1991). It is surprising, therefore, that there has been so little attention dir-
Regional stresses in the heart ected toward quantifying possible regional variations in the biomechanical properties of cardiac tissue. Hence, our findings provide new information on the regional mechanical behavior of normal noncontracting canine myocardium, information that is essential for subsequent studies on regional disease-induced changes in properties. In particular, the present results, together with our previous findings on ventricular epicardium, suggest that the biaxial mechanical properties of the noncontracting canine heart are heterogeneous. The behavior of ventricular epicardium is significantly different from that of myocardium, which in turn appears to vary transmurally within the LV free wall; outer-wall myocardium is stiffer than that in the underlying midwall (p=O.O6), and may be stiffer than inner-wall myocardium as well. Because of the nonlinear behavior of cardiac tissue, even slight differences in material properties may have significant effects on predicted wall stress. Thus, this issue of myocardial heterogeneity merits further, careful attention.
Acknowledgments-We thank Mr John Downs and MS Kristi Pier for their technical expertise. Partial financial support from NIH grants HL 41130 and HL 33621 are gratefully acknowledged. Portions of this work were performed in partial fulfillment of the M.S. degree by VPN at UMBC.
REFERENCES Buccino, R. A., Harris, E., Spann, J. F. and Sonnenblick, E. H. (1969) Response of myocardial connective tissue to development of experimental hypertrophy. Am. J. Physiol. 216, 425-428. Demer, L. L. and Yin, F. C. P. (1983) Passive biaxial mechanical properties of isolated canine myocardium. J. Physiol. (London) 339,615-630. Downs, J., Halperin, H. R., Humphrey, J. D. and Yin, F. C. P. (1990) An improved video based computer tracking system for soft biomaterials testing. IEEE Trans biomed. Engng 31,903-907.
Glass, L., Hunter, P. J. and McCulloch, A. D. (1991) Theory of Heart. Springer, New York. Guccione, J. M., McCulloch, A. D. and Waldman, L. K. (1991) Passive material properties of intact ventricular myocardium determined from a cylindrical model. ASME J. biomech. Engng 113, 42-55. Huisman, R. M., Sipkema, P., Westerhof, N. and Elzinga, G. (1980) Comparison of models to calculate ventricular wall force. Med. biol. Engng Comput. 18, 133-144. Humphrey, J. D., Strumpf, R. K. and Yin, F. C. P. (199Oa) Biaxial mechanical behavior of excised epicardium. Am. J. Physioi. 259, H 101-H108. Humphrey, J. D., Strumpf, R. K. and Yin, F. C. P. (1990b) Determination of a constitutive relation for passive myocardium: I. A new functional form. ASME J. biomech. Engng 112, 333-339. Humphrey, J. D., Strumpf, R. K. and Yin, F. C. P. (199Oc) Determination of a constitutive relation for passive myocardium: II. Parameter estimation. ASME J. biomech. Engng 112, 340-346. Humphrey, J. D., Strumpf, R. K. and Yin, F. C. P. (1992)
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A constitutive theory for biomembranes: application to epicardial mechanics. ASME J. biomech. Engng 114, 461-466.
Humphrey, J. D. and Yin, F. C. P. (1989a) Constitutive relations and finite deformations of passive cardiac tissue. II. Stress analysis in the left ventricle. Circ. Res. 65, 805-817. Humphrey, J. D. and Yin, F. C. P. (1989b) Biomechanical experiments on excised myocardium: theoretical considerations. J. Biomechanics 22, 377-383. Janz, R. F. and Grimm, A. F. (1972) Finite element model for the mechanical behavior of the left ventricle. Circ. Res. 30, 244-252. Janz, R. F. and Grimm, A. F. (1973) Deformation of the diastolic left ventricle. I. Nonlinear elastic effects. Biophys. J. 13,689-704.
Janz R. F. and Waldron, R. J. (1976) Some implications of a constant fiber stress hypothesis in the diastolic left ventricle. Bull. Math. Biol. 38, 401-413. Kent, R. S., Carew, T. E., LeWinter, M. M. and Covell, J. W. (1978) Comparison of left ventricular free wall and septal diastolic compliance in the dog. Am. J. Physiol. 234, H392-H398. Lew, W. Y. W. and LeWinter, M. M. (1986) Regional comparison of midwall segment and area shortening in the canine left ventricle. C&c. Res. 58, 678-691. Line. D.. Rankin. J. S.. Edwards. C. H.. McHale. P. A. and Anderson, R. W. (1979) Regional diastolic mechanics of the left ventricle in the conscious dog. Am. J. Physiol. 236, H323-H330. Mirsky, I. (1970) Effects of anisotropy and nonhomogeneity on left ventricular stresses in the intact heart. Bull. Math. Biophys. 32, 197-210. Mirsky, I. (1979) Elastic properties of the myocardium: a quantitative approach with physiologic and clinical applications. In Handbook ofPhysiology (Edited by Berne, R. M., Sperelakis, N. and Geigen, S. R.), pp. 497-531. American Physiologic Society, Bethesda, MD. Moriarty, T. F. (1980) The law of Laplace: its limitations as a relation for diastolic pressure, volume or wall stress of the left ventricle. Circ. Res. 46, 321-331. Novak, V. P. (1992) Regional mechanical properties of passive myocardium. M.S. thesis, University of Maryland. Baltimore. Nielsen, P. M. F., LeGrice, I. J., Smaill, B. H. and Hunter, P. J. (1991) Mathematical model of geometry and fibrous structure of the heart. Am. J. Physiol. 260, H1365-H1378. Omens. J. H. and Fung, Y. C. (1990) Residual strain in rat left ventricle. Circ. Res. 66, 37-45. Przyklenk, K., Connelly, C. M., McLaughlin, R. J., Kloner. R. A. and Apstein, C. S. (1987) Effect of myocyte necrosis on strength, strain and stiffness of isolated myocardial strips. Am. Heart J. 114, 134991359. Shacklock, A. (1987) Biaxial testing of cardiac tissue (1987) M.S. thesis, University of Auckland, New Zealand. Smaill, B. H. and Hunter, P. J. (1991) Structure and function of the diastolic heart. In Theory ofHeart (Edited by Glass, L., Hunter, P. J. and McCulloch, A. D.). Springer, New York. Villarreal, F. J., Waldman, L. K. and Lew, W. Y. W. (1988) Technique for measuring regional two-dimensional finite strains in canine left ventricle. Circ. Res. 62, 711-721. Waidman, L. K., Fung, Y. C. and Covell, J. W. (1985) Transmural myocardial deformation in the canine left ventricle: normal in uioo three-dimensional strains. Circ. Res. 57, 152-163. Yin, F. C. P. (1981) Ventricular wall stress. Circ. Res. 49, 829-842. Yin, F. C. P., Strumpf, R. K., Chew, P. H. and Zeger, S. L. (1987) Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading. J. Biomechnnics 20, 577-599.
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APPENDIX For simplicity, consider an equibiaxial test without shear. That is, let ,& =& =1 and K~= K*=O. Thus, even with a linear splay of muscle fibers through the thickness [see equations (4) and (S)] u (=A) and 1, (= 21’ + l/12’) are independent of @, thereby allowing equation (6) to be integrated easily, namely
+w,L
1’
(Al)
I
644
[
1 sin 2mT-sin 2Q ;+ 4(%-W
[
1 sin 2DT-sin 2mB 24(%-W
+w,n
Now, consider the following special cases. Case I: In the limit as C++O and @a-O, equations (Al) and (A2) become (fl,)=fll=2W~(~2-l/124)+W,rl, (t2*)=tZZ=2W~(r22-l/14),
(A3)
which, as we expected, are the same as those in Humphrey et al. (1990b) for the case of no fiber splay and N=(l, 0,O). Similarly, when QT-+ n/2 and @a-+ n/2, one obtains the same results except that t,, is the stress in the fiber direction and t, 1 the stress in the cross-fiber direction. Case II: As 4 -+ n/2 and (Pa+ 0, equations (Al) and (A2) become (tll)=(t22)=2W~(~Z-l/124)+t~IZ,
(A4)
which is again intuitively obvious. That is, biaxial stresses should be the same in an equibiaxial test when the muscle fiber splay changes linearly from 0 to 7r/2. Case Ill: As mT+ j? and @a+ - /?,equations (Al) and (A2) yield
W Thus, as fi increases from 0 to n/2. the difference between (t,,) and (tz2) becomes less and less. That is, differences between the predominant fiber and cross-fiber stresses (e.g. Figs 1 and 2) become less and less with increased symmetric fiber splays even though the true fiber direction is stiffer. This important observation must be remembered when interpreting anisotropy from plots of experimental data.