engineering mechanics for successive states in canine left

Engineering Mechanics for Successive Statesin Canine Left Ventricular Myocardium

II . FIBER ANGLE AND SARCOMERE LENGTH

By Daniel D. Streeter, Jr., and William T. Hanna

ABSTRACTThe relations between end-diastolic (D) and end-systolic (S) fiber angles (a) and

sarcomere lengths (s) have been previously studied at different sites in canine left ven-tricular myocardium. However, no postulates have been advanced for predicting aand s in successive states of the ejection cycle (D or S) or at different sites in onestate when the semimajor (Z) and semiminor (R) axes of the wall surfaces for succes-sive states and the fiber orientations and sarcomere lengths at one site in one state areknown. In this study, the myocardial fibers were regarded as the matrix of a myocar-dial continuum: they are prisoners of the heart wall and must comply with the move-ments of the wall. Using the same values as in the preceding article, the wall wastreated as a nested set of truncated ellipsoidal shells of revolution with shell volumespreserved from D to S. Both confocal and nonconfocal configurations were analyzed.In each shell, the fibers were assumed to follow a "helical" path with a constant ad-vance in each turn about the Z axis (the simplest possible path). The results of thisassumption were compatible with previously reported values of a and s measured atvarious sites in the left ventricular free wall in states D and S. These results suggesta postulate for the heart wall: in the beating heart, each muscle fiber changes directionand length uniquely as the wall changes shape.

KEY WORDS dog heart fiber orientation helical pathellipsoid of revolution confocal ellipses nested set of shellsleft ventricular wall shape truncated ellipsoid grid spacing

• A precise knowledge of the myocardial fiberstructure of the left ventricular wall (1) isimportant to the physiologist, because it leads to theestablishment of laws relating possible fiber motion(2) to the overall motion of the wall. This papershows that changes in individual left ventricularfiber angle (3-5) and sarcomere length (5) can berelated to the change in the shape of the leftventricular wall throughout the cardiac cycle. Thereare no statistically significant data supporting thechange in fiber angle (4, 5 ) , although there aredata supporting the change in sarcomere length(5) . The modeling of left ventricular wall shaperests on the assumptions previously given for anested set of truncated ellipsoids of revolution

(1) .The fibers in left ventricular myocardium can be

From the Department of Pathology, University ofWashington School of Medicine, Seattle, Washington 98195,and Battelle Columbus Laboratories, Columbus, Ohio43201.

This research was supported by the Washington StateHeart Association, by U. S. Public Health Service Grant 5P01-HL 13517 from the National Heart and Lung Institute,and by Battelle Institute.

Received May 7, 1973. Accepted for publicationOctober 10, 1973.

likened to those in a wicker basket (Fig. 1). If awicker basket made of soft fibers is loaded heavily,the fibers change in two ways: they run moresteeply and they stretch out to some extent. Thus,the fibers change their direction and their length.Since they are prisoners of the basket wall, theirchanges must reflect the changes in the shape of thebasket.

The left ventricle of the heart, which is a moresophisticated structure, can be visualized as anested set of fiber shells (2) (Fig. 2). On each ofthese shells, the muscle fiber makes a spiral andadjacent fibers are parallel to each other. Proceed-ing sequentially from shell to shell, the angle of thespiral changes very gradually (3, 4) (Fig. 3). Ablock removed from the full width of the wall andsliced as in Figure 4 reveals that the fiberorientation resembles that of the opened blades of aJapanese fan. On each blade, the fibers follow theline of the blade, but each fiber is randomlytethered to its neighbor within the blade and fromblade to blade (Fig. 5). Thus the whole wall is atethered fiber matrix. In this tethered network, thefibers are restricted to a lateral sliding or rollingmotion (4), and no fiber can act independently.Each is postulated to move as its neighbors allow it

656 Circulation Research. Vol. XXX11I, December 197}

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CANINE LV FIBER ANGLE AND SARCOMERE LENGTH 657

FIGURE 1

Wicker basket made of sop fibers. Left: Unloaded. Right:Loaded heavily. Note that fiber angle, a, steepens and thatfibers lengthen.

to move, because the heart is a single fiber con-tinuum (2). Beyond the endocardium, the fibers,as trabeculae and papillary muscle elements, arefreed of lateral constrains (6). Elswhere, they aresubject to the constraints of the wall (5).

FIGURE 2

Nested set of fiber shells. On each of the shells the fibermakes a spiral, and adjacent fibers are approximately parallelto each other.

Circulation Research, Vol. XXXIII, December 197}

If a fiber on one of the many nested shells makingup the heart wall is isolated (Fig. 6), a grid ofimaginary lines of latitude running circumferential-ly and lines of longitude running orthogonally tothem can be printed on the shell surface. For agiven segment of a fiber, e.g., a fibril one sarcomere

ENDOCARDIUM

MID-WALL

- 1 0 0 0 M -

EPICARDIUMFIGURE 3

Photomicrographs in horizontal sequence showing myocardialfiber orientation vertically through the free wall of a dilatedcanine left ventricle near the root of the anterior papillarymuscle. The sections are parallel to the epicardial plane.Their left-hand edge is the natural left side of the blockexcised from the wall. Fiber angle is +60° at the endo-cardium and runs through 0° at the midwall to —75° atthe epicardium.


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658 STREETER, HANNA

FIGURE 4

Left ventricular wall, modeled as a nested set of fiber shells,with the excised block carved to reveal a fiber orientationlike that of blades of an opened Japanese fan.

in length, fenced in with these lines so that itsdirection and length is demarcated, the fiberorientation is indicated by the fiber angle (a)measured in the plane of the shell from a line oflatitude, the sarcomere length (s), the circumferen-tial grid spacing (a), and the longitudinal gridspacing (b). When the shape of the heart changesduring the cardiac cycle, as from end-diastole (D)to end-systole (S), both the length and the angle ofthe fiber segment are postulated to change accord-ingly (Fig. 7).

Note that the fiber segment is fenced in by thesame gridwork in each state. Since the fiber isconstrained, the relative change in a and s can bemeasured by the relative change in a and b. Incomparing state S with state D, observe thatchanges in a and b are accompanied by distinct,measurable changes in the semiminor (R) andsemimajor (Z) axis dimensions of the shell but thatthe longitudinal location factor of point P, f = z/Zremains constant. Thus, changes in a and s at pointP can be predicted by changes in the semiaxisdimensions of the left ventricular wall.

A postulate for the heart wall is implied by thisconstruction: in the beating heart, each muscle fiberchanges direction and length uniquely as the wallchanges shape. This postulate was tested usingpreviously published canine data.

MethodsEQUATIONS

The relation between the fiber angle, a, and the griddimensions enclosing it (Fig. 6) is

FIGURE S

Opened Japanese fan modeling the tethering of fibers withina blade and between neighboring blades. Fibers are gen-erally parallel to their neighbors.

tan ctjli)h = bj1t)JajtiiL, (1)

where the subscript j denotes the latitude (e.g., thelongitudinal angle of a line of constant latitude), thesubscript i denotes the shell in the fiber continuumbetween the two wall surfaces, and the subscript Ldenotes successive end-states (D or S) in the cardiaccycle or a sequence of counting numbers representingshorter time intervals in the cardiac cycle. Thesarcomere length is given by

SU,L - °i,*,t/cos ajjj,. (2)When the left ventricular wall changes shape from

state D to state S, the new fiber angle (Fig. 7) is givenby

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tan aj ff = tan a

The new sarcomere length is

iDlaUiiS). (3)

In a previous paper of this series (1), thecircumferential grid spacing between two lines oflongitude at 8 + A0/2 and 6 — A0/2 was given by

AcJ-4ii = R^Aflcos <)>), (5)

and the longitudinal grid spacing between two parallelcircles at <j>j 4- A ^ / 2 and <>;- — A<f>j/2. was given by

^djjj, = Z^A(^j(l — eifi* s*n */*;)*• (®)

The ellipsoidal shell eccentricity, eiL, is

Assume the value of unity for A6 and A<j> and assumethe same length units for Ac, Ad, R, and Z. The fenced-in fiber grid dimensions are then related to these griddimensions as follows:

where SOjj and d<f>j,i are the latitudinal and longi-tudinal angles of arc, respectively, for the grid.These angles of arc are generally different for successive,values of i in adjoining shells and ; at each latitude inthe left ventricular wall but invariant during thecardiac cycle.

Thus, the new fiber angle, ctj^s, can be related to

the old angle by the semiaxis dimensions in each state L(D and S).

tan a M S = tane^f

(Zi > s/Zi l 0)(10)

Also the new sarcomere length, SjiH, can be related tothe old.

>s oijii>s), ( 1 1 )

where the longitudinal location factor, /;-, is given by

= sin' (pj, ' (12)

and eiL is given by Eq. 7. This relation applies only toa fixed point in the wall at two instants of the cardiaccycle.

It is possible to relate the fiber angles at two differentpoints on a shell at one instant of the cardiac cycle. Toachieve this relation requires a postulate, such as that ofSallin (7), that the fibers follow the simplest possiblepath on the shell, namely, a "helical" path with aconstant advance in each turn about the Z axis. Such apath constitutes a spiral whose radial projection onto aconcentric cylinder forms a circular helix of constantpitch. The quantity that characterizes such a fiber,iltj,, is the total angle of wrap of the fiber on the shellabout the Z axis from equator to apex. The constantfiber angle on a cylinder of radius RiLl is an iJt at theequator of the ellipsoid whose semiaxes are Rj>£, andZj r • its relation to these dimensions is

tan (13)

FIGURE 6

Muscle fiber on one of the nested shells that arbitrarily de-fines a "fiber layer" in the heart wall. Lines of latitude andlongitude fence in a segment of the fiber, demarcating itsdirection and length. Fiber direction is indicated hy theangle, a, measured from a line of latitude in the plane ofthe shell; the length of the fiber segment is indicated by s.

Circulation Raettcb, Vol. XXX1I1, December 1973

End diostole (D) End systole (S )FIGURE 7

Fenced-in muscle fiber segment at arbitrary fixed point, P,on an ellipsoidal myocardial shell in two successive statesof the cardiac cycle, showing change in fiber direction, a,and length s, during contraction. Circumferential and longi-tudinal grid dimensions, a and b, respectively, define fiberangle and length in each state. R and Z = semiminor andsemimafor axis dimensions of shell, and z = distance fromequatorial plane to point P. The longitudinal location fac-tor, fj = z/Z, which defines latitude, is the same in eachstate. Note that the fiber is imprisoned in the gridwork—itcannot move independently of the shell. This constructionimplies a new postulate (see text).


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At other longitudes, < , the fiber angle on the ellipsoidis given by

tan aiA,L = ZirJ(R^iX^cos <^cos y,-,i,J, (14)

where y^tj, is the conical taper (3) found in therelation

tan yhiiL = R ^ tan d>jlZiL. (15)

The ratio of Eq. 14 to Eq. 13 gives a^i,^ in terms of

tan aj^L — tan ao,i,L[l+(Ri,L tan <f>,[ZirLy]ycos if,,. (16)

Although the nomenclature is that of the left ventricle,these equations are equally appropriate for otherhelically wound, tethered fiber structures, if theirconfiguration is retained as part or all of an ellipsoid ofrevolution at each instant of time.

APPLICATION OF EQUATIONS TO THE ANALYSIS OF DATA

Eq.16 was used with the measured equatorial fiberangle, ao,<,n> associated with semiaxis data R ^ andZ(// at L=D or S (2) to give the diastolic fiberangle, a{ j D, through the wall at five given latitudes(f = 0.6,'0,5, 0.7, 0.9, and 0.99). Eq. 10 was usedwith each calculated value of OLJXD

a n d the sameassociated semiaxis data to give a,%s through the wallat the same five given latitudes. No theory is proposedfor the diastolic distribution of sarcomere lengths,

Sj i D, across the wall of the left ventricle arrested atend-diastole. Eq. 11 was used with a value of sjinmeasured at the equator (2), which was assumed to bethe same (for convenience) for all values of /'(latitudes), to give the systolic sarcomere length,Sjjg, which relates to the calculated fiber angle,a, i L, and the pair of semiaxis dimensions, R( L andZiL (L-D or S).

Result's and DiscussionFIBER ANGLES

In Figure 8, calculated left ventricular fiberangles are seen at three different latitudes as afunction of percent wall thickness in representativecanine ventricles, modeled as nonconfocally nested,truncated ellipsoids of revolution (1) at end-diastole (A) and end-systole (B). The characteris-tic sigmoid shape of the curve at the equator(/y = 0.0) steepens, straightens, and then reversesits shape as the apex is approached. Superimposedare data from previously studied T-shaped speci-mens from the left ventricular free wall (8), the T-top representing fy = 0.0 and the T-leg representingfj — 0.5-0.9. Only the diastolic fiber angles from theT-top and the semiaxis dimensions of the wall ineach state were fed into Eqs. 10 and 16 to producethe plots. Note that the sigmoid shape of each T-leg

DIASTOLE SYSTOLE

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20 40 60 80Percent of wall thickness

100EPI

FIGURE 8

Fiber angles through nonconfocal left ventricular wall thickness from endocardium to epicar-dium at various latitudes, 0, 7, and 9. Calculated fiber angles at t} are 0.0 at the equator, and0.7 and 0.9 near the apex. T-TOP and T-LEG: Measured fiber angles in T-shaped specimenwith top extending along equator and leg extending from equator to apex of free wall; ±SE isindicated. A: At end-diastole. B: At end-systole. All calculations are based on T-top data inA. Fiber angles through confocal left ventricular wall are similar.

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curve shows an increase in steepness and straight-ness relative to that for the T-top. Furthermore, thediastolic T-leg data closely match the calculatedcurve (Fig. 8A) in the latitude given by f — 0.7.Also note that the T-top fiber angle data in systolediffer from the corresponding calculated fiberangles (Fig. 8B, f, = 0.0) by a constant angle. Thisangle represents a bending of the Z axis (4) whichcannot be mimicked by the ellipsoid-of-revolutionmodel (1). A similar observation applies to thesystolic T-leg fiber angles and the calculated fiberangles (Fig. 8B, f, = 0.7) except in the endocardialquartile of the wall. However, these angles are theresult of two geometric transformations (Eqs. 10and 16) which each impose strong geometricalassumptions (1) on the way the fiber angleschange. It is indeed remarkable that the calculatedand the measured data are as consistent asindicated. These results roughly support not onlythe imprisoned-fiber concept but also the wallmodel as a truncated nested set of ellipsoids ofrevolution. Confocal and nonconfocal wall modelsdo not significantly differ in the fiber anglecalculations.

Fiber Angle Changes from End-Diastole to End-Systole.—Although previous studies have generally

not succeeded in verifying any significant change infiber angle between the end-states D and S (4, 5),small changes are shown in the calculated values inFigure 9 for the equator (A) and for a latitudeclose to the apex (/; = 0.9) (B). Observe that thefibers in each shell increase in steepness more in theendocardial half of the wall and move more epi-cardially relative to wall thickness (to preservethe volume of each shell during contraction). Thenet effect is to enhance the angle change endocar-dially and to cancel it epicardially. The calculatedchange is greater at the equator than it is near theapex. It is interesting to note, however, that the T-leg data nearest to the apex do show a significantfiber angle change, which is atypical (4). Theseobservations suggest that a limitation on thevalidity of the wall model must be recognized nearthe apex, perhaps at fa = 0.9.

Limitations of Wall Model—In Figure 10, thecalculated fiber angle distribution across the wall isshown for latitudes that asymptotically approachthe apex. For f, = 0.99, a calculated curve appearsthat has no experimental support (4). It indicatesthat, in the limit, all apical fiber angles are either+90° or —90°. Gross visual evidence points to theexistence of a translucent apical vortex of fibers

8 0

40 0

EPI

0 20 40 ,Percent wall thickness

8 0 IOO

-20

- 4 0

- 6 0

L-80

ENPO

20 40Percent s>

80

sol

4 0 <

•Si

20

60 80wall thickness

100

- 6 0

- 8 0

FIGURE 9

Calculated change in -fiber angles, end-diastole (D) to end-systole (S), through nonconfocalleft ventricular wall. Solid circle = D and open circle = S. A: At f. = 0.0 (equator). B: Atfj = 0.9 (near apex). Fiber angle changes in confocal wall are similar.

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662 STREETER, HANNA

80-

0 20 40 60 80 100ENDO Percent of wall thickness EPI

FIGURE 10

Calculated fiber angles at end-diastole in nonconfocal leftventricular wall at carious latitudes, 0, 5, 7, 9, N; f} = 0.0(equator), 0.5, 0.7, 0.9, 0.99 (near apex), respectively.Fiber angles in confocal wall are similar.

making an endocardial infundibulum, which is mostapparent in ventricles that are in rigor (1). Sincethe wall model does not account for this infundibu-lum, it is clearly invalid at the apex.

A more adequate representation of fiber orienta-tion at the apex may be that of an apex diaphragm,similar to the iris diaphragm of a camera ormicroscope (9). Since the wall is a tethered fibermatrix which, when "unwound," reveals a surfaceon which fibers are preferentially torn (3),conceptual ventricular modeling requires that thearchitectural integrity of a region be recognized.The ellipsoidal shell model, where valid, requiresrecognition that the shells are not like onion layersthat can be peeled apart but are indeed artificialand arbitrary, because the fiber continuum does notenvisage any shells. Similarly, an iris diaphragmmodel of the apex implies recognition that the

number of diaphragmatic leaves that are carved outis arbitrary and that the leaves are indeed artificial,because the fiber continuum does not envisage theirseparateness. Viewed in this way, a better concep-tualization of apical structure may be obtained.

SARCOMERE LENGTHS

In Figure 11, calculated diastolic left ventricularsarcomere lengths at end-systole are seen as afunction of position in the wall at the equator of therepresentative, nonconfocal ventricle discussed inthe preceding section. These values were obtainedfrom three assumed distributions of sarcomerelengths representing end-diastole, which are alsoshown. In one distribution, sarcomere length wasassumed to be constant across the wall at 2.07/u. and11 mm Hg, based on midwall measurements of thesame five left ventricles in state D that were usedfor fiber angle measurements (2, 10). The other twodistributions were based on measurements acrossthe wall at the quartile points at filling pressures of6 and 12 mm Hg (11). Also included are foursarcomere length distributions obtained by Hort(5) across the wall for two states of left ventriculardistention and two of left ventricular contraction.The overall pattern of the calculated distributions isconsonant with the data available. The leastpredicted movement of sarcomeres during contrac-tion is at the epicardium; the greatest is at 25% ofthe wall thickness measured from the endocardium.Predicted fiber shortening at midwall and at theendocardium are about the same.

There is some diversity in the shape of the threesarcomere length distributions assumed for thediastolic state. The constant distribution across thewall in state D, based on only one midwall site ofmeasurement, causes the distribution in state S toshorten in a most pronounced way at the firstquartile point through the wall. The two distribu-tions of Yoran et al. (11) provide for a less abruptchange in the sarcomere length distribution acrossthe wall. Note that the last two curves for the Dand S states are almost mirror images of each other,a seemingly more efficient arrangement.

It is easy to see why the greatest sarcomeremovement occurs at the first quartile point throughthe wall. Two factors must be considered: (1) amidwall fiber is oriented more circumferentiallythan is one at the endocardium and will participatemore fully in the predominantly circumferentialcontraction that normally characterizes ejection,and (2) a point at midwall lies further from the Zaxis than does one at the endocardium and will

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CANINE LY FIBER ANGLE AND SARCOMERE LENGTH 663

2.4-

0 20 40 60 80ENDO Percent of wall thickness

FIGURE IT

100EPI

Sarcomere length through left ventricular wall at equatorfrom endocardium to epicardium. Lines A', B', and C areassumed diastolic distributions through the wall. Line A' isderived from diastolic data of Sonnenblick et al. (10). LinesB' and C are derived from diastolic data of Yoran et al.(11) at 6 mm Hg and 12 mm Hg, respectively. Yoran's dataat 2 mm Hg (•) and at 20 mm Hg (?) are also given. Line Ais the calculated nonconfocal systolic distribution in the wallfrom the data of Sonnenblick et al. (10). Lines B and Care the calculated nonconfocal systolic distributions in thewall from the data of Yoran et al. (11) when diastolic pres-sure is 6 mm Hg and 12 mm Hg, respectively. The dottedlines represent the data of Hort (solid triangle = stronglydilated heart, solid circle = dilated heart, open diamondsmedium filling, solid diamond = rigor). The cross-hatchedarea indicates the range of changes in sarcomere lengthpredicted from the 6-tnm Hg diastolic data of Yoran et al.(11). Since systolic sarcomere lengths must fall betweenmedium filling and rigor values, we note that the calculatedvalues are not unrealistic. Note that the overall pattern ofchanges in- sarcomere length is least at the epicardium andmaximum at 25% of wall thickness according to the calcu-lated values. This finding can easily be explained (see text).

Circulation Research, Vol. XXXIll, December 1973

participate less fully in the radially inward motionof the wall during ejection. These two factors whichare functions of radial position in the wall followopposite courses. Maximum fiber shortening is atrade-off that must occur somewhere between thetwo points. Hence, this pattern of shortening is areasonable one that gives food for thought.

Limitations of Wall Model—In Figure 12, thecalculated sarcomere length distribution across boththe confocal and the nonconfocal wall is shown forlatitudes that asymptotically approach the apex.There is little difference between the two ways ofrepresenting the ellipsoids. Again, at ft = 0.99, acalculated curve appears which has no experimentalsupport. It reveals that an endocardial sarcomere atthe end of normal ejection has been shortened to alength which is less than that of the myosin filament

2.1 I2.0-

1.9-

$ 1-8-

£

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1.6-

1.5-

1.40

ENDO20 40 60 80Percent of wall thickness

100EPI

FIGURE 12

Calculated sarcomere length at end-systole through left ven-tricular wall from endocardium to epicardium for variouslatitudes, comparing confocal (broken line) with nonconfocal(solid line) wall models. Latitudes = 0, 5, 7, 9, N; fj — 0.0(equator) and 0.S, 0.7, 0.9, 0.99 (near apex), respectively.A uniform sarcomere length of 2.07/J (JO) is assumed at D.Maximum shortening occurs at 25% of wall thickness at alllatitudes except those near the apex. See text for discussionof limitations.


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664 STREETER, HANNA

and less than Hort's endocardial sarcomere in rigor(5); this pattern deviates peculiarly from that atother latitudes. As with the fiber angles calculatedat /, = 0.99, the sarcomere length equations appearto be pushing the wall model beyond its limits ofcredibility. In fact, the earlier speculation that thislimit might even be at fj = 0.9 receives supportfrom the reversed trend seen in the sarcomerelength curve for that latitude.

ADVANTAGES OF WALL MODEL

The left ventricular wall, modeled as a nested setof confocal or nonconfocal ellipsoidal shells, hasbeen shown to approximate the real left ventricularwall in the region that extends from the equator towithin 90% of the Z distance (measured from theequatorial plane to the apex). This approximationpermits numerical estimates to be made of a fiberangle and sarcomere length in the left ventricularwall, anywhere except at the apex, when (1) thesevalues are known at another instant, (2) thesemiaxes are defined for the truncated ellipsoid ofrevolution on which the fiber lies at each instant,and (3) the latitude of the wall site is defined.

The Postulate.—The ellipsoid model serves asa convenient, compatible (1) vehicle for testing apostulate for the heart wall: in the beating heart,each muscle fiber changes direction and lengthuniquely as the wall changes shape. While unique-ness has not been demonstrated, the postulatedchanges in fiber angle and sarcomere length havebeen demonstrated and the changes in sarcomerelength have been verified at five points in the leftventricular wall. The differences in measured fiberangles at end-diastole and end-systole have not yetbeen shown to be significant (4, 5). It is suggestedthat such demonstrations be extended to cases inwhich more opportunities exist for testing foruniqueness. It is also suggested that other modelsbe developed for testing this uniqueness to

permit apical fiber orientations and sarcomerelengths to be demonstrated and tested.

AcknowledgmentWe wish to thank Eddie R. Hamerly for his Calcomp

Graphics assistance, Alison Ross for her editorship, andSuzanne Corbett for typing the manuscript.

References1. STHEETER, D.D., JR., AND HANNA, W.T.: Engineering

mechanics for successive states in canine leftventricular myocardium: I. Cavity and wall geome-try. Circ Res 33:639-655, 1973.

2. STREETER, D.D. JR., VAISHNAV, R.N., PATEL, D.J.,

SPOTNITZ, H.M., Ross, J., JR., AND SONNENBLICK,

E.H.: Stress distribution in the canine left ventricle indiastole and systole. Biophys J 10:345-363, 1970.

3. STHEETEH, D.D., JR., AND BASSETT, D.L.: Engineering

analysis of myocardial fiber orientation in pig's leftventricle in systole. Anat Rec 155:503-511, 1966.

4. STHEETER, D.D., JR., SPOTNITZ, H.M., PATEL, D.J.,

Ross, J., JR., AND SONNENBLICK, E.H.: Fiber orienta-tion in the canine left ventricle during diastole andsystole. Circ Res 24:339-347, 1969.

5. HORT, W.: Makroskopische und mikrometrisehe Unter-suchungen am Myokard verschieden stark gefiillterlinker Kammern. Virchows Arch [Pathol Anat]333:523-564, 1960.

6. GAY, W.A., AND JOHNSON, E.A.: Anatomical evaluationof the myocardial length-tension diagram. Circ Res21:33-43, 1967.

7. SALLIN, E.A.: Fiber orientation and ejection fraction inthe human left ventricle. Biophys J 9:954-964,1969.

8. STBEETER, D.D., JR., AND SPOTNTTZ, H.M.: Orientation

of myocardial fibers: Dog. In Respiration and Circu-lation, edited by P. L. Airman and D. S. Dittmer.Bethesda, FASEB, 1970, 229-230.

9. TORRENT GUASP, F.: The Cardiac Muscle. Madrid,Torroba, 1973.

10. SONNENBLICK, E.H., Ross, J., JR., COVELL, J.W.,

SPOTNITZ, H.M., AND SPIRO, D.: UJtrastructure of the

heart in systole and diastole: Changes in sarcomerelength. Circ Res 21:423-431, 1967.

11. YORAN, C , COVELL, J.W., AND Ross, J., JR.: Structuralbasis for the ascending limb of left ventricularfunction. Circ Res 32:297-303, 1973.

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DANIEL D. STREETER, JR. and WILLIAM T. HANNAFiber Angle and Sarcomere Length

Engineering Mechanics for Successive States in Canine Left Ventricular Myocardium: II.

Print ISSN: 0009-7330. Online ISSN: 1524-4571 Copyright © 1973 American Heart Association, Inc. All rights reserved.is published by the American Heart Association, 7272 Greenville Avenue, Dallas, TX 75231Circulation Research

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