Relation Between Transmural Deformation and Local Myofiber Direction in Canine Left Ventricle

Transcription

1 55 Relatin Between Transmural Defrmatin and Lcal Myfiber Directin in Canine Left Ventricle Lewis K. Waldman, David Nsan, Francisc Villarreal, and James W. Cvell T determine the relatin between lcal myfiber anatmy and lcal defrmatin in the wall f the left ventricle, bth three-dimensinal transmural defrmatin and myfiber rientatin were examined in the anterir free wall f seven canine left ventricles. Defrmatin was measured by imaging clumns f implanted radipaque markers with high-speed, biplane cineradigraphy (16 mm, 12 frames/sec). Hearts were fixed at end diastle and sectined parallel t the lcal epicardial tangent plane t determine the transmural distributin f fiber directins at the site f strain measurement. The principal directin f defrmatin assciated with the greatest shrtening was cmpared with the lcal fiber directin in the uter (21 ±8% f the wall thickness frm the epicardium) and inner (65 ±9%) halves f the wall. Althugh the fiber directin varied substantially with depth frm the epicardium, the principal directin did nt. In the uter half f the wall, fiber directin averaged 8 ± 24, while the principal directin averaged 33 ±24 frm circumferential (cunterclckwise angles are psitive). In the inner half, fiber directin averaged 69 ±1, while the principal directin averaged 22 ±21. Therefre, while fiber and principal directins were nt substantially different hi the uter half, the greatest shrtening ccurred rthgnally t the fiber directin hi the inner half. Nrmal and shear strains measured in a cardiac crdinate system (circumferential, lngitudinal, and radial crdinates) were rtated (transfrmed) t "fiber" crdinates hi bth halves f the wall. In the uter half, nrmal strains bserved in the fiber (.9 ±.4) and crss-fiber (-.4 ±.4) directins were nt significantly different (paired t test, p<.5). In the inner half, mre than twice as much strain ccurred in the crss-fiber (-.17 ±.3) than in the fiber directin (.6 ±.6). Mrever, the nly shear strain that remained substantial after transfrmatin was transverse shear in the plane f the fiber and radial crdinates. These results suggest that bth rerientatin and crss-sectinal shape changes f myfibers r the interstitium may cntribute t the large wall thickenings bserved during cntractin, particularly in the inner hah* f the ventricular wall. (Circulatin Research 1988;63:55-562) The way in which myfibers lying at different depths in the heart wall and having different rientatins interact during cntractin is nt knwn. It is well knwn, hwever, that there is an extensive cllagenus netwrk surrunding the mycyte and cllagen struts between mycytes. 12 Mrever, the directin f the myfibers, which varies cntinuusly with depth spanning mre than 1 acrss the anterir free wall f the canine left ventricle, 34 is neither altered by large changes in ventricular mass r shape 3 nr affected greatly by Frm the Departments f Medicine and Biengineering, University f Califrnia, San Dieg, La Jlla, Califrnia. Supprted in part by Natinal Heart, Lung, and Bld Institute grants PHS-HL , HL-17682, and HL Address fr crrespndence: Lewis K. Waldman, PhD, Department f Medicine, M-13-J, University f Califrnia, San Dieg, La Jlla, CA Received February 5, 1987; accepted March 25, cntractin (i.e., myfiber directin changes abut 2 in the transitin frm diastle t systle 34 ). Despite the fact that myfiber directin varies acrss the wall and that sarcmeres wuld be expected t shrten nly alng their axes, there is accumulating evidence that the principal directins f defrmatin vary little frm epicardium t endcardium. 6-7 These results imply that there must be substantial shrtening defrmatin that is nt aligned with the axes f the lcal myfibers. There is additinal evidence that supprts this pssibility; fr example, several investigatrs using piezelectric crystal dimensin gauges and ther techniques have shwn defrmatin away frm the lcal fiber directin, particularly near the epicardium Mrever, there is evidence that the directin f greatest shrtening (the principal directin assciated with the mst negative principal strain) can vary substan-

2 Waldman et al Left Ventricular Defrmatin and Fiber Orientatin 551 tially with time during cntractin 6 and that the principal directins f defrmatin can be changed by an easily reversible interventin such as altering the ventricular activatin sequence, 6 which shuld nt influence fiber directin. Hwever, few studies have examined fiber directin at the same site at which strain is measured, and nne have examined strains and fiber directins at mre than ne transmural psitin acrss this thick-walled structure. Finally, few investigatrs have explited techniques f cntinuum mechanics that prvide the capability t cmpute tw- and three-dimensinal principal defrmatins frm strain data measured in arbitrary crdinate systems. Thus, it was the bjective f the present study t test the hypthesis that the principal directins f defrmatin in the anterir wall f the canine left ventricle differ frm the lcal myfiber directin. Accrdingly, we examined three-dimensinal transmural finite defrmatin with high-speed cineradigraphy in seven pen-chest, anesthetized dgs and cmpared the principal directins with the myfiber directins determined in these same hearts fixed at end diastle. The results indicate that there is substantial shrtening defrmatin perpendicular t the lcal fiber directin, particularly in the inner half f the left ventricular wall. Materials and Methds Seven mngrel dgs weighing frm 18 t 25 kg were anesthetized with pentbarbital (3 mg/kg) and given supplemental dses when necessary. The animals were intubated, and respiratin was maintained with a Harvard respiratry pump. A midline sterntmy and a bilateral thractmy at the fifth intercstal space were perfrmed, and the heart was supprted in a pericardial cradle. A micrmanmeter (mdel p5, Knigsberg) was inserted int the left ventricle thrugh a stab wund in the apex. The manmeter was calibrated against a pigtail catheter inserted int the left ventricle frm the iliac artery and cnnected t a Guld-Statham P23db gauge (Cleveland, Ohi). Then, the catheter was withdrawn t the artic rt t mnitr artic pressure. T cmplete the preparatin, three clumns f frm fur t six lead r tungsten carbide beads (.d., mm) were implanted in the anterir free wall f the left ventricle as described previusly. 7 In brief, the marker clumns were implanted with a trcar psitined with the aid f a plastic template sutured t the epicardium. This template cnsisted f three hles arranged in a right triangle with 8-mm legs (hyptenuse, mm). Then, the template was remved, and reference system marker beads were sewn t the epicardium at the bifurcatin f the circumflex and anterir descending crnary arteries, at the left ventricular apex, and ver each clumn f beads. Care was taken t psitin the animal in the x-ray fields s as t avid verlap f the beads during cntractin. Lead II f the electrcardigram, camera shutter crrelatin marks, and artic and ventricular pressures were recrded n an scillgraphic recrder (mdel Mark 2, Guld). With respiratin suspended at end-expiratin, biplane cineradigraphy (16 mm, 12 frames/sec) was used t determine the bead psitins. Then, as described belw, the heart was prepared fr fixatin at the same end-diastlic pressure bserved during strain measurement. Anatmic Studies The azygs vein was ligated, and sutures were psitined arund the cavae, the hila, and the ascending arta. The left main crnary artery was dissected free, and a ligature was passed arund it. A catheter was placed in the left atrium fr the infusin f saline t adjust the filling pressure f the arrested heart. The ligatures cntrlling inflw were tied dwn, and filling pressure was adjusted t the in viv value in the hypxic, arrested heart. Then, a Gregg cannula was psitined in the left main crnary artery, secured with a suture, and a pwer injectr (mdel Injectr 1, Crdis) was used t inject slutin t fix the in situ heart. 916 Tw fixative slutins (Biuns slutin and buffered frmalin) 917 were used in the present study with n apparent differences in the results. The measurement f lcal mycardial fiber directin in this labratry has been described in detail previusly. 59 In brief, a skewer was passed thrugh the fixed, silastic-filled heart frm the apical dimple t the mitral aspect f the artic valve, and the heart was placed in a special cutting jig. A reference plane was cut perpendicular t this apex-base axis (Figure 1A). Then, a blck f tissue (1 cm 2 in the epicardial tangent plane) was remved that included the mycardial markers. T facilitate embedding and remval f the beads, the blcks were cut in halves r thirds with sectins parallel t the epicardial tangent plane. The epicardial edge was marked with a suture, and the circumferential reference edge was marked with a grve. After careful dissectin t remve the beads, ne r tw transmural blcks were imbedded in paraffin and sectined parallel t the epicardial tangent plane. Sectins were munted at.5-mm intervals acrss the wall. Measurements f fiber directin were made at three r mre sites in each slide, and the results were averaged. Finite Defrmatin The technique fr determinatin f lcal transmural finite strains has been described in detail recently. 67 Sme defrmatin data but n fiber directin measurements r transfrmed strains (see belw) have been reprted previusly fr five f the seven animals studied (i.e., fur f these five dgs were used in ur first transmural defrmatin study 7 in which the statistical significance f the principal directin data was nt examined, and the remaining dg was the first f eight animals used in ur secnd transmural defrmatin study 6 ). In brief, sets f fur nncplanar markers demarcating small tetra-

3 552 Circulatin Research Vl 63, N 3, September ' FIGURE 1. Schematic diagram f the left ventricle shwing the site at which defrmatins and fiber rientatin were measured. Panel A: Ventricular site in the anterir free wall and the cardiac crdinate system t which the strains are referred (C, circumferential; L, lngitudinal; and R, radial). Panel B: Psitins f three clumns f radipaque markers. Panel C: Fur markers demarcating a typical subepicardial tetrahedrn used in strain analysis are shwn. hedra f muscle were used t cmpute symmetric finite strain tensrs at several transmural lcatins (Figures IB and 1C). After cmputing threedimensinal nrmal and shear strains referred t the cardiac crdinate system at end diastle, 614 an algebraic eigenvalue prblem was slved t calculate the three principal strains and their directins with respect t the end-diastlic reference crdinates. There were always several pssible tetrahedra available fr analysis. An attempt was made t select tetrahedra that met the fllwing criteria: three beads at similar depth (within 1.5 mm) defining the base f a tetrahedrn, height f the furth bead within mm frm the base, and eigenvlumes (prducts f principal stretches) f the tetrahedra at end systle f In the present study, single cntractins were selected at lw levels f left ventricular enddiastlic pressure fr strain analysis. The timing f end systle was defined by first determining the end-systlic artic pressure frm the fluid-filled artic rt catheter. T crrect fr the phase lag inherent in this system, the timing f end systle was determined at this pressure n the left ventricular micrmanmeter tracing. The micrmanmeter and artic pressure prfiles were carefully matched with this catheter in the left ventricle befre it was withdrawn int the artic rt. T determine the relatin between lcal finite strains and myfiber directin, the end-systlic principal directin assciated with the first principal strain (greatest shrtening) in a given tetrahedrn and the myfiber directin in the same heart at the same relative depth (percent wall thickness) frm the epicardium as the centrid f that tetrahedrn were cmpared. Bth the fiber directin and the first principal directin were identified similarly as an angle frm the circumferential directin (cunterclckwise, psitive) in the epicardial tangent plane and were designated respectively as the fiber angle and the in-plane angle. Because fiber directin infrmatin was available at.5-mm intervals, linear interplatin between pints was perfrmed where necessary. The reference system used fr the descriptin f strains was smewhat arbitrary; the principal defrmatins were, f curse, independent f it, but the nrmal and shear strains were nt. In deference t the shell-like gemetry f the left ventricle and the presence f circumferential fibers near midwall, typically the crdinate directins in planes parallel t the epicardial tangent plane have been chsen t be circumferential and lngitudinal ("cardiac" crdinates). Hwever, because we knew the lcal fiber directin at the centrid f each tetrahedrn, it was pssible t transfrm (rtate) the strain tensr t a new crdinate system in which the new "surface" crdinates were in the lcal fiber directin and perpendicular t it ("fiber" crdinates). The inplane rtatin is represented by the familiar rtatin matrix: cs -sin e sin cs e e l which des nt affect the radial crdinate and its crrespnding nrmal strain, E 33. The strain variables fr strains ccurring in the epicardial tangent plane (in-plane strains) are circumferential strain (E M ), lngitudinal strain (E^), and in-plane shear (E l2 ) in cardiac crdinates. After the in-plane rtatin described abve, they becme fiber strain (Eg), crss-fiber strain (E«), and transfrmed in-plane shear (Ef C ). The remaining strains ccurring in planes perpendicular t the epicardial tangent plane are the transverse shears E 3 and E^ in cardiac crdinates r Ef, and E c3 in fiber crdinates. The radial strain is E 33. Nrmal strains, as is the custm in cntinuum mechanics, are the extensinal strains (i.e., E M, E n, E 33, Eff, and E^). The principal strains are defined in increasing rder as E, (greatest shrtening), E 2, and E 3 (greatest thickening). Results Studies were perfrmed in seven mngrel dgs with an average left ventricular weight f 112 ± 3 g

4 Waldman et al Left Ventricular Defrmatin and Fiber Orientatin 553 TABLE 1. End-Systlic Nrmal Strains Outer Mean±SD Depth (%) Depth (nun) ± 1.6 E,, ±.5 E ±.4 Strains E ±.15 Err ±.4 Ecc ±.4 Inner Mean±SD ± ± ± ± ± ± ±.3 Outer, tetrahedra were chsen with centrids at transmural psitins lcated between midwall and epicardium; inner, centrid psitins between midwall and endcardium; En, circumferential strain; E22, lngitudinal strain; E 33, radial strain; Efj and E^, nrmal strains after rtatin f the strain tensr (see "Materials and Methds") in the fiber directin and perpendicular t it (crss-fiber), respectively. (mean±sd). Heart rate and left ventricular pressure averaged 12 ±24 beats/min and 125 ±19 mm Hg (peak) and 4.7 ± 1.5 mm Hg (end diastle). The filling pressures at the time f fixatin were within 1 mm Hg f the end-diastlic pressure at which strains were measured in each animal. The lcatin f the left ventricular site chsen t sample bth strains and fiber directins was n the anterir free wall f the left ventricle abut halfway between the left anterir descending crnary artery and the lateral wall. Differences in the crnary anatmy f each dg resulted in sme variatin in the selectin f ventricular sites; hwever, the centrid f the three epicardial beads was always between 1 and 2 cm frm the left anterir descending artery and averaged 66 ± 6% f the distance frm base t apex. In each animal, data frm a single cntractin were analyzed. Tetrahedra were selected fr an analysis f the end-systlic principal directin f greatest shrtening and ther strain variables (see "Materials and Methds") at several sites acrss the wall. An end-diastlic reference cnfiguratin was used fr all strain calculatins. A ttal f 26 tetrahedra (16 frm the uter half and 1 frm the inner half f the ventricular wall) were analyzed frm the seven dgs. In cases in which mre than ne tetrahedrn was available in either half (tw t fur tetrahedra), the strain data were averaged in each half. Then, mean values btained in the tw halves fr each f the seven animals were averaged. The strain data frm individual animals and the verall averages and standard deviatins are shwn in Tables 1,2, and 3. At least tw tetrahedra were selected in each animal fr the analysis f strains during the curse f an entire cntractin starting and ending at end diastle. Figure 2 shws a typical example f the time curse f left ventricular pressure (Panel A), finite strains (Panel B), and the directin f greatest shrtening (Panel C) during a single cntractin in a subendcardial tetrahedrn (centrid at 11.1 mm deep r 79% f the wall thickness at end diastle). The circumferential strain, E u, reaches apprximately -.12 (negative nrmal and principal strains indicate shrtening) at end systle (25 msec after end diastle). The first principal strain E,, which we have defined as the greatest negative strain, is f greater magnitude at end systle. Hwever, because the assciated principal directin is largely circumferential in this example, E M and E, have very similar time curses. The angle between the principal directin assciated with E, and the circumferential directin (inplane angle, Panel C) is typically quite negative near end diastle and changes rapidly tward the circumferential directin during cntractin. Mst tetrahedra lcated in the inner half f the wall demnstrate a similar rtatin tward the circumferential axis that cntinues thrugh systle and well int diastle. Figure 3 shws the in-plane angle at three different levels (Panels A, B, and C) acrss the wall in the same experiment illustrated in Figure 2. The ejectin phase f the cntractin determined frm the crssver f left ventricular and artic pressure prfiles (see "Materials and Methds") starts at 1 msec frm end diastle, and the principal directin during ejectin is quite cnstant acrss the wall

5 554 Circulatin Research Vl 63, N 3, September 1988 TABLE 2. End-Systlic Shear Strains Strains Ere ±.3 Outer Mean±SD E, E,, ±.4 E Ef, ±.6 Ec Inner Mean ± SD ± ± ± ± ± Outer, tetrahedra were chsen with centrids at transmural psitins lcated between midwall and epicardium; inner, centrid psitins between midwall and endcardium; E, 2, shear strain in the epicardial tangent plane; Ei 3, transverse shear strain in the plane f the circumferential and radial crdinates; E23, transverse shear strain in the plane f the lngitudinal and radial crdinates; Ej c, En, and Ec3 are the shear strains after rtatin f the strain tensr int the fiber directin (see "Materials and Methds"). ranging between -4 and -6 frm circumferential. There is mre scatter in the principal directin data during isvlumic cntractin and diastasis when the strains are small and the reslutin f the eigenvectrs is either pr r indeterminate because f the multiplicity f vanishing eigenvalues. The end-systlic in-plane angle averaged - 33 ± 24 in the uter half and - 22 ± 21 in the inner half f the wall, indicating a transmural unifrmity f principal directins similar t but smewhat mre circumferential than that bserved in a previus study. 6 Other strains shwed the anticipated increase in magnitude with depth frm the epicardium acrss the wall. 7 Fiber directin was examined in a blck f tissue that included at least tw f the three clumns f radipaque markers in fur f the seven dgs. In the remaining three animals, tissue preparatin was unsatisfactry, and a secnd blck was cut adjacent t the bead set. Data frm a typical animal are shwn in Figure 4. The epicardial fiber angle was - 25 and prgressed t +11 frm circumferential at the endcardium. Linear regressins were perfrmed n the fiber angle data as a functin f percent depth in each animal, and the cefficients were averaged yielding a slpe f 1.56 ±.14 /% depth and an intercept f -49± 15 (fiber angle at the epicardium). The data differed frm that f Streeter 4 in that the hrizntal intercept f the curve (depth at which fibers were circumferential) ccurred at a mre epicardial site averaging 31 ± 8% f the wall thickness frm the epicardium. Furthermre, ur fiber angle curves were mre linear with little f the curvilinear (cubic) prperty bserved n the average by Streeter 4 that suggested a prepnderance f circumferential fibers in the middle third f the wall and mre rapidly varying directins in the inner and uter thirds. Hwever, a careful examinatin f differences in the transmural distributin f fiber directins between "T tp" and "T leg" ventricular sites examined by Streeter 4 rather than his "averaged" data revealed very similar trends as cmpared with what we fund at sites that were prbably all T leg. Mrever, ur sites were significantly mre septal (5 mm clser) than Streeter's sites. Figure 4 shws the relatin between the end-systlic principal directin f greatest shrtening (as indicated by the in-plane angle) and fiber directin as a functin f depth frm the epicardium in the same experiment illustrated in Figures 2 and 3. There is an increasing divergence between principal and fiber directins with depth frm the epicardium leading t a substantial difference between the end-systlic principal directin (-22 ±21 ) and the myfiber directin ( +69 ± 1 ) in the inner half f the wall. Figure 5 shws average in-plane strain data frm all experiments. Here, strains present in planes parallel t the epicardial tangent plane (circumferential, lngitudinal, and in-plane shear strains) have been transfrmed (rtated in the plane) s that the nrmal strains are nw riented in the fiber directin and perpendicular t it (see "Materials and Methds"). A paired t test was used t establish statistical significance (p<.5). Near the epicardium (uter half, Panel A), n significant difference was fund between strain in the fiber directin

6 Waldman et al Left Ventricular Defrmatin and Fiber Orientatin 555 TABLE 3. Outer Mean±SD Fiber Directin and End-Systlic Principal Defrmatins Fiber angle (degrees) ±24 In-plane angle (degrees) ±24 E, ±.2 Strains E ±.3 E, ±.16 Inner Mean±SD ± ± ± ± ±.15 Outer, tetrahedra were chsen with centrids at transmural psitins lcated between midwall and epicardium; inner, centrid psitins between midwall and endcardium; fiber and in-plane angles are measured with circumferential being and cunterclckwise angles being psitive. Ei, first principal strain (greatest shrtening); E 2, secnd principal strain; E 3, third principal strain (greatest lengthening r thickening). (-.9 ±.4) and perpendicular t it (-.4 ±.4). In the inner half f the wall (Figure 5B), there is significantly mre strain perpendicular t the fibers (-.17 ±.3) than in the lcal fiber directin (-.6 ±.6). End-systlic nrmal strains ccurring in the uter and inner halves f the ventricular wall in cardiac crdinates and after rtatin t fiber crdinates (see "Materials and Methds") are shwn in Table 1. Data frm individual animals as well as averages and standard deviatins fr all seven animals are presented. Accmpanying shear strains are shwn in Table 2. In cardiac crdinates, small but cnsistent levels f psitive in-plane shear (E )2 ) were bserved acrss the wall, while substantial transverse shears (E^ and E23) existed nly in the inner half. After transfrmatin, mst shearing cmpnents were small except fr a substantial level f transverse shear (E n =.9 ±.5) in the inner half f the wall in the crdinate plane defined by the fiber and radial directins. Fiber angles and endsystlic in-plane angles and principal strains ccurring in the uter and inner halves f the ventricular wall are given in Table 3. Nte the presence f substantial shrtening defrmatins (E, and E2) alng tw f the principal directins in the inner half f the wall. Discussin The present study is the first t examine fiber rientatin and three-dimensinal finite defrmatin simultaneusly in the canine left ventricle. The results clearly indicate that there is a unifrm rientatin f the principal directin f greatest shrtening acrss the free wall f the left ventricle while fiber directin changes substantially. Recent studies using radigraphic and ther techniques indicate the presence f shrtening defrmatins nt in the fiber directin. Thus, Ingels et al, 18 Arts et al, 1 Dieudnne, 12 Fentn et al, 13 and Meier et al 14 have shwn defrmatin in several directins at a single site. Mre recently, several investigatrs using piezelectric dimensin gauge systems that have substantially greater tempral and spatial reslutin than ther techniques have shwn that there is defrmatin perpendicular t the lcal fibers in the epicardium and midwall Fr example, the data f Lew and LeWinter 11 shw substantial lngitudinal shrtening in the midwall averaging nearly half f the circumferential shrtening bserved there. The present study and recent studies frm this labratry have been the first t systematically examine three-dimensinal transmural defrmatins at a single site. 67 These studies clearly shw that the principal directin f defrmatin assciated with the greatest shrtening des nt cincide with the fiber directin in the inner half f the anterir free wall site examined. The present study implies that there must be either substantial rearrangement f myfibers during cntractin r large shape changes in the crsssectins f mycytes r the interstitium. Several lines f evidence supprt the cncept that the principal directins f defrmatin may nt be determined cmpletely by the lcal fiber directin. The extensive cllagen netwrk between mycytes

7 556 Circulatin Research Vl 63, N 3, September 1988 X E a> D </) <n CD ^- _l C Stra *«* CD s > c Ia. c circumferential % A. i B. /^^-^ C A * A 14 A ** A 1 O A ~ '* \ OO A \ - ' C. OOffi ** O O a > a, \ -, A «V V.. V^" " V =b ^*> "» Time (msec) FIGURE 2. P//s f typical time series data frm ne cardiac cntractin in ne experiment are shwn. ED, end diastle; ES, end systle. Panel A: Left ventricular pressure. Panel B: Tw cmpnents f the finite strains ccurring in the inner half f the ventricular wall are shwn (i.e., the circumferential strain [E,,] and the first principal strain [Ei, greatest shrtening]). Nte that E, is greater than E t, during much f the cardiac cycle. Panel C: Orientatin f the principal directin assciated with E, is indicated by the in-plane angle it makes with the circumferential crdinate directin (cunterclckwise angles are psitive with circumferential at ). cursing in different directins wuld be expected t reduce shrtening in the directin f individual mycytes by lading them in ther directins. That there is shrtening defrmatin rthgnal t the lcal fiber directin that is greater than fiber shrtening in the inner half f the ventricular wall suggests that there is sme srt f gemetric rearrangement f mycytes during cntractin. This pssibility has been prpsed by Sptnitz et al 2 frm the results f studies n passive dilatin f rat hearts. Figure 6 is adapted frm their wrk and shws three ptential cnfiguratins fr the free wall. In Panel A, the simple cncept that cylindrical fibers shrten alng their axes and that their crsssectins lengthen symmetrically is illustrated. Clearly, bth lcal and glbal measurements refute this pssibility because neither psitive lngitudinal strains nr base-t-apex lengthening have been bserved with cntractin. Mrever, wall thickening and ejectin fractins realized frm such a mechanism are t small. Panel B demnstrates the rerientatin cncept f Sptnitz et al. 2 In diastle, the fibers are arranged at an angle t each ther, and with cntractin they mve relative t ne anther and are reriented mre perpendicular t the epicardial tangent plane. Thus, Sptnitz et al 2 have fund that there is a direct crrelatin between the increase in wall thickness and the radial fiber cunt in fixed, diastlic hearts. That the nly shearing cmpnent that remains substantial after transfrmatin t fiber crdinates is the transverse shear in the plane f the fiber and radial directins in the inner half f the wall supprts the cntentin f Sptnitz et al, 2 wh have calculated that an angle change f abut 2 during cntractin is necessary t accunt fr thickening. Hwever, the magnitude f this transverse shear when expressed as an angle change suggests that it is nly partly respnsible fr the large thickenings bserved. Cnverting the average shear (En =.9, Table 2) t an angle, the infinitesimal shear angle is cmputed as: a = atan(2e f3 )=1.2 while the finite shear angle is actually smewhat smaller because f the presence f the large radial strain: a = asin (2Ef 3 N\+lE a VI + 2E33) = 8.5 The angle change indicated in Figure 6B exaggerates this cmpnent fr illustrative purpses. The ther pssibility, that there may be cell shape changes, is shwn in Figure 6C. Here, cell r fiber bundle crss-sectins defrm s that material circles in diastle becme ellipses with cntractin. The same effect culd be achieved by changes in shape f the interstitial spaces. 21 Regardless f the lcal mechanisms that result in the bserved defrmatins, the large wall thickenings prduced during cntractin in the left ventricle are nt accunted fr by simple cnsideratins f the Pissn effect, that is, vlume-preserving fiber shrtenings d nt result in lengthening (thickening) alng the tw directins mutually rthgnal t the fiber axes (Figure 6A). Rather, the mycardium tends t underg tw-dimensinal shrtening, that is, negative lngitudinal and secnd principal strains are substantial when cmpared with circum-

8 Waldman et al Left Ventricular Defrmatin and Fiber Orientatin EPI 18 r 9 a Fiber dlrtclln In-ptvn angtm. -9- DD <h ^ circumfttmn tl&l -18tf 8 a 18 9 < CD B. c V MID ENDO **~ss'a' Time (msec) FIGURE 3. Plts f the rientatin f the principal directin assciated with the greatest shrtening (E,) as indicated by the in-plane angle (see Figure 2C) are shwn at three depths in ne experiment during ne cardiac cycle. Panel A: EPI, epicardium 5% f the wall thickness frm the epicardium. Panel B: MID, midwall 45%. Panel C: ENDO, endcardium 79%. Nte the similarity f the in-plane angle at end systle (ES) and the relative stability f the in-plane angle during ejectin at all three depths. ED, end diastle. ferential and first principal strains, particularly in the inner half f the ventricular wall. In this way, the lengthening (thickening) that must accmpany shrtening in a vlume-preserving (incmpressible) defrmatin is unidirectinal, prviding a pwerful mechanism fr ejectin. The ne-dimensinal fractinal shrtening measured by snmicrmeters des nt have the capability t accunt fr shearing defrmatin (either in the epicardial tangent plane [in-plane shear] r transverse t it). Therefre, the frmer methd can nly allude t maximal shrtening because principal strains and directins are nt available. Fr example, the study f Freeman et al 9 did nt measure true principal strain but instead equated the strain measured in the circumferential directin t maximal shrtening. The effect f in-plane Percent Depth FIGURE 4. Plt f transmural trend in the rientatin f fiber and principal directins (as indicated by the in-plane angle) at end systle in ne experiment. Endcardium, 1%. Observe the transmural unifrmity f the in-plane angle and the divergence between in-plane angle and fiber directin with increasing depth frm the epicardium. shear n the interpretatin f cardiac defrmatin data has been discussed previusly by Prinzen et al 22 in respnse t the Freeman study and in previus studies frm bth labratries Our current results crrbrate these earlier bservatins. A tw-dimensinal methd with a triangular arrangement f radipaque markers 14 r snmicrmeters 15 r the mre cmprehensive threedimensinal methd develped in ur labratry 7 can be used t cmpute tw- r three-dimensinal symmetric strain tensrs, thus allwing fr the cmputatin f principal data. As these tensr methds have cnsistently shwn, the principal directin assciated with the greatest systlic shrtening is almst always riented alng directins that crrespnd t negative Streeter fiber angles because f the presence f small but cnsistent in-plane shearing defrmatin acrss the wall at the ventricular site examined. Therefre, it is unlikely that circumferential shrtening at midwall is indicative f peak shrtening. The examples described in ur first study f finite defrmatin in mycardium 7 suggested that peak shrtening at midwall and deeper may ccur at angles between -1 and -6 (Figure 7 f Waldman et al 7 ). The mre cmprehensive data appearing in a mre recent study 6 again shwed that this principal directin ften has little transmural variatin (Table 1 f Waldman et al 6 : the epicardial in-plane angle was -56 ±27, and the endcardial angle was -46 ±24 ) and certainly varies much less than the fiber directins measured by Streeter. 34 The results f the present study are a direct manifestatin f the relative transmural unifrmity f this principal directin. A clue t this cnsistent finding was already visible in the data f Freeman et al. 9 Referring t Figure 6 f Freeman et al, 9 bserve that ver a range f angles (f the axis f the crystal pairs) f 2-3 frm the lcal fiber directin, a substantial number f data pints are at 1% maximal shrt-

9 558 Circulatin Research Vl 63, N 3, September 1988 OUTER 4. B -.4 B L INNER c a >- FlGURE 5. Bar charts f average in-plane nrmal and shear strains at end systle in the uter ("Panel A, 22±8% f the depth frm the epicardium) and inner (Panel B, 65±9%) halves f the ventricular wall after rtatin t "fiber" crdinates (see "Materials and Methds"). Errr bars, standard deviatins. Fiber, nrmal strain in the fiber directin; 9 Deg, nrmal strain in the crss-fiber directin; Shear, in-plane shear remaining after rtatin. Observe that mst f the extensinal strain in the inner half f the wall is perpendicular t the lcal fibers, while the remaining in-plane shear is virtually zer. ening. If Freeman et al had shearing data, sme f these pints wuld have been greater than 1%. We further illustrate the differences between single and multidimensinal techniques in Figure 7. Here, we selected a very typical set f shrtening and thickening strains that we (and thers, albeit nt simultaneusly) have bserved at lcatins near mid wall and deeper (-.15 circumferential strain, -.7 lngitudinal strain, and +.34 radial strain). The questin we ask is, given these nrmal strains, what happens t the rati f fiber strain t crssfiber strain as we vary the presumed fiber directin thrugh psitive Streeter angles? This prcedure uses the in-plane rtatin described in the present study t cmpute the fiber and crss-fiber strains as we vary fiber directin thrugh angles that are representative f thse measured at midwall and FIGURE 6. Schematic diagrams indicating three pssible cnfiguratins f the ventricular wall at end diastle (ED) and end systle (ES). h, wall thickness at ED; h', wall thickness at ES. Panel A suggests a simple realizatin f the Pissn effect in which fiber shrtening results in symmetrical lengthening f the fiber crss-sectins. Here, the resulting wall thickening is inadequate t accunt fr bserved wall thickenings and ejectin fractins. Panel B indicates rerientatin f myfibers r interstitial spaces surrunding them during cntractin. Panel C indicates that crss-sectinal shape changes f myfibers r the interstitium may ccur. Here, unlike in Panel A, material crss-sectins defrm nnsymmetricalty with the bserved defrmatin field in which tw-dimensinal shrtening cntributes t the large wall thickenings. deeper. Equivalent results culd be btained with a simple applicatin f Mhr's circle. In additin, what is the effect f increasing amunts f in-plane shear n this rati? Figure 7 shws that withut any shear (as assumed in the study by Freeman et al 9 ) fiber strain dminates crss-fiber strain ver a range f fiber angles f 45. Hwever, with small amunts f psitive shear, the rati falls t unity mre rapidly. Fr example, with the nrmal strains specified and in-plane shear f.4, similar t ur average data, the rati is abut ne third r less (E cc /E ff >3) ver a range f fiber angles frm abut + 6 t +8. If in-plane shear is.6, E cc /E ff >3 ver a range f fiber angles frm abut + 42 t This is a wide range f pssible fiber angles. Thus, the result that crss-fiber strain dminates fiber strain ver a range f midwall t subendcardial fiber angles is an inevitable cnsequence f the presence f psitive in-plane shear. It is an interesting crllary that with increasing amunts f shear, the range f psitive fiber angles ver which fiber strain dminates crss-fiber strain becmes

10 Waldman et al Left Ventricular Defrmatin and Fiber Orientatin 559 FIGURE 7. Plt f rati f fiber strain (E ff ) t crss-fiber strain (E^) as a functin f fiber angle at different levels f in-plane shear strain (En). Refer t "Discussin." ANGLE (degrees) increasingly narrw. Fr example, with a shear f.4, fiber strain is greater than crss-fiber strain ver a range f fiber angles frm t Fr a shear f.6, this range is nly frm t 17. These results are nt affected by the presence f transverse shearing defrmatins. The difference in transmural psitins (depth) in the Freeman et al 9 data and the data frm the present study culd als make a significant difference in the magnitude f crss-fiber strain. Althugh Freeman et al d nt state the average depth f the measurements and these hearts are n lnger available, it seems reasnable t assume that the piezelectric crystals were implanted 5 mm frm the epicardium and t use the average wall thickness frm the present study (12.75 mm). The Freeman et al data wuld then be at 39% f the distance frm the epicardial surface t the endcardium. In the present study, the average depth at which the fiber directin was was 3.7 ±7.6% (±SD). Because the Freeman et al data were measured at a similar site, it seems likely that the fibers were near circumferential in that study. Frm the analysis shwn in Figure 7, it seems very likely that fiber strain wuld always exceed crss-fiber strain at the level examined in the earlier study. 9 On the ther hand, the average fiber directin in the inner half f the wall in the present study (at mre than twice the depth at which fibers were circumferential n average) was + 69, and crss-fiber strain exceeded fiber strain at these depths frm the epicardium. With regard t the variatin f fiber shrtening between the uter half (-.9 ±.4) and the inner half ( ), there was n significant difference between fiber shrtening at these transmural psitins in the anterir free wall site examined. The current data prvide further evidence that myfibers acrss the free wall f the left ventricle tend t wrk tgether as a unit, but the mechanism that cnstrains fibers f widely varying rientatin t shrten cllinearly is difficult t discern withut a 9. lcal measure f stress r a cmplete understanding f the three-dimensinal material prperties. In additin t the mechanisms suggested earlier, ur bservatins raise several questins. First, is it pssible that the cllagen netwrk itself is anistrpic, that is, that tethering acts mainly in planes parallel t the epicardial tangent plane cnstraining the first principal directin but allwing fr substantial freedm fr radial defrmatin? Secnd, is frce generatin in the midwall and subepicardium greater than in the inner wall? Fr example, if stresses have mre transmural unifrmity than the magnitudes f the crrespnding strains, as has been suggested n theretical grunds by arguments invking either trsin 24 r residual stress mechanisms, then the frces acting in an uter half f substantially greater vlume than the inner half f the ventricular wall culd be mechanically dminant. Mrever, is it cnceivable that there are transmural variatins in material prperties r that neighbring fiber ppulatins lcated mre basally than ur ventricular site and having mre circumferential fibers may interact with the site we have chsen? These questins await further study. There are a number f ptential surces f errr in this study. First, we are lking at nly ne site n the ventricular wall, which has been chsen because it is free f ptential abrupt changes in fiber directin because f papillary muscles. Hwever, in the apex f the left ventricle where fiber directin is changing rapidly and at the base f the heart where valve rings cmplicate the anatmy, different results are pssible. Nevertheless, even thugh we carefully avided using either fiber r strain data that invlved the anterir papillary muscle directly, its ptential influence cannt be ignred. Secnd, we have cmpared the end-systlic principal directin (directin f greatest shrtening) with the fiber directin determined in the same heart arrested in diastle. If there are substantial changes in the fiber directin during systle, then the discrepancy

11 56 Circulatin Research Vl 63, N 3, September 1988 between fiber and principal directins culd be an artifact f the end-diastlic reference cnfiguratin. Hwever, Streeter et al 34 have examined fiber directin in hearts fixed during cntractin and have shwn that the fiber directin changes abut 2 frm its directin in the diastlic cnfiguratin at any transmural psitin. Of critical imprtance t these data is that the transmural variatin f fiber directins is nt diminished by cntractin supprting the bserved difference between principal directin and fiber directin. Here, an additinal assumptin cncerns the timing f the systlic fixes perfrmed by Streeter et al. 3-4 Because the duratin f fixatin is several hundred millisecnds, these are nt necessarily end-systlic fixes. There are several surces f variability that cntribute t the wide standard deviatins shwn in Tables 1-3. First, the depth f the measurements varies substantially frm animal t animal, ranging frm 14.4% t 31.8% fr the uter half and frm 5% t 74.4% fr the inner half. Because strain magnitudes vary with depth, this will add t the variability in the magnitudes f the strains. Hwever, the directin f greatest shrtening des nt vary much acrss the wall in each animal but des tend t vary substantially frm animal t animal. Fr example, the difference in this principal directin between endcardium and epicardium varied frm.75 t 24 in all the animals, while the endcardial in-plane angle varied between - 3 and -55 in all animals. Sme f this latter variability is likely attributable t the hemdynamic status f the animal. There is preliminary evidence frm this labratry that increases in end-diastlic pressure shift the end-systlic in-plane angle tward the circumferential directin. 15 Cncern abut errrs in the measured fiber directins prmpted us t examine ptential errrs at varius stages f the fixatin prcess. Bth fixatives used in the present study were administered by arterial perfusin, and bth resulted in a rapid (1-secnd) fixatin. Bth agents are knwn t alter the vlume f the tissue. Buffered frmalin prduces a 15-2% decrease in vlume, whereas glutaraldehyde changes vlume very little (±5%) althugh the fixatin is smewhat slwer. The errrs ccurring during the fixatin prcess have been addressed befre. 16 In this study, there were n apparent changes in ventricular shape r crss sectin after rapid fixatin with glutaraldehyde. T increase ur cnfidence that there were n majr alteratins in ventricular shape with the buffered frmalin used in the present study, we have repeated the analysis in tw dgs with glutaraldehyde and fund n substantial differences. The ptential fr errr that wuld affect the results f these studies (and Streeter's initial wrk) is much greater during the dehydratin phase. During this phase, there is a large reductin in wall vlume. Hwever, because the blck f tissue cntaining the beads is remved befre the dehydratin, the ptential exists fr either nnunifrm shrinkage r substantial shearing distrtin, which culd influence the results. Accrdingly, the tw additinal hearts fixed with glutaraldehyde were dehydrated in alchl ver a perid f 1 week s that bth ttal wall vlume and finite strains (and, thus, lcal muscle vlume) culd be measured during the dehydratin prcedure befre cutting the blck. Fr this purpse, the reference cnfiguratin was that f the fixed heart befre dehydratin. In the tw hearts, there was substantial shrinkage f the ttal wall vlume (frm 73 t 42 ml and frm 78 t 44 ml). Hwever, this appeared t be evenly distributed acrss the wall with a 4-5% lss f muscle vlume in bth epicardial and endcardial tetrahedra. Mrever, the shear strains ccurring frm beginning t end f the dehydratin were t small t distrt measured fiber directins substantially. In these tw additinal hearts, the fiber angle-depth relatins were very similar t thse bserved in the remaining seven animals. The variability f fiber angles within sectins is discussed in the study f Freeman et al 9 and was nt measured systematically in the present study. Three angles were measured frm each sectin, and the range was usually less than 15. In ne f the tw hearts described abve, the maximum range f fiber angles in each sectin averaged 1 ±9 in 132 samples. The variability adjacent t a sectin can be determined frm the data f Streeter 3 t be between 1 and 2 in either the apex-base directin (leg) r circumferential directin (base) alng the T-shaped area studied by Dr. Streeter. Neither the variability in fiber directin within sectins nr that at adjacent sites affects the average transmural shape f the fiber directin-depth relatin and, thus, des nt qualitatively affect the cnclusins f this study. The influence f deviatins f fiber rientatin frm the epicardial tangent plane n ventricular defrmatin is nt knwn. It is nt clear whether the transverse shears we measure are attributable t the small "traverse" angle described by Streeter 3 r t ther pssible mechanisms, such as the deviatin f lcal shape frm axisymmetry r the influence f neighbring ventricular sites. Regardless f the mechanism, we d indeed bserve psitive transverse shear strains in the inner half f the ventricular wall as shwn in Table 2. Mrever, if the transverse shear f.9 ±.5 (En f Table 2) is a secnd means by which inner-half fibers cntribute t the ttal defrmatin besides nrmal strain in the fiber directin (Eg), this wuld help explain their verall functin. The calculatins f Wiseman et al 25 indicate that the assumptin f unifrm strain within the tetrahedral vlume results in errrs attributable t the transmural variatins in strain cmpnents in thickwalled bdies. These errrs were demnstrated by chsing different tetrahedral gemetries within the walls f cylinders and spheres, prescribing defrmatins, and cmputing the exact strains at the cen-

12 Waldman et al Left Ventricular Defrmatin and Fiber Orientatin 561 trids f the tetrahedra as well as the experimental strains with a fur-pint calculatin (which assumes unifrmity). Spheres and cylinders were subjected t negative inflatin and trsin. In additin, negative extensins were impsed n cylinders. We have repeated these calculatins fr the range f tetrahedral gemetries used in ur experiments. As described in detail in ur first study n finite defrmatin in mycardium 7 and repeated here fr clarity in the subsequent discussin, subendcardial tetrahedra are chsen with three pints apprximately in the same plane and the furth pint mm clser t the epicardium, and subepicardial tetrahedra have three pints in a plane near the epicardium and the furth mm deeper. Midwall tetrahedra have been chsen tipping either up r dwn. When a subendcardial tetrahedrn is arranged as described, the three pints near the endcardium tend t weight the shrtening calculatins tward the endcardium where shrtening is greater, while the radial leg f the tetrahedrn weights the thickening calculatins t be slightly clser t the epicardium (representative f the midpint f the radial leg rather than the centrid f the tetrahedrn). Thus, shrtening is verestimated and thickening is underestimated with the result that the vlume appears t decrease. The subepicardial arrangement des just the ppsite, biasing the calculatin tward the epicardium. Here, shrtening is underestimated and thickening is verestimated, resulting in apparent vlume increases. Frtunately, ur tetrahedral gemetries result in substantially smaller errrs than many f thse chsen by ur clleagues in Baltimre. The reasn is that ur intermarker and interclumn spacing are such that the radial span f the tetrahedra is ften cnsiderably less than 4 mm and the circumferential span is between 5 and 7 mm. The calculatins indicate that it is the interactin between errrs intrduced by finite dimensins in bth the epicardial tangent plane and radial directins that tends t amplify these errrs. Many representative calculatins fr bth the sphere and cylinder have been made. The results indicate that the strain errrs and vlume verestimatin near the epicardium are exceedingly small. In the subendcardium, the strain errrs including deviatins in principal directins are quite small. Nevertheless, vlume underestimatins f 5-1% ccur and shuld be kept in mind when interpreting radial strains and lcal vlume changes. Generally, the errrs were fund t be slightly greater fr the spherical mdel. Depending n the chice f tetrahedral gemetry, the errrs in the circumferential, lngitudinal, radial, first transverse shear (E 13 ), secnd transverse shear (E^), first principal, secnd principal, and third principal strains ranged frm -.8 t -.23, -.11 t-.29, -.33 t -.88,.17 t.46, -.25 t.19, -.12 t -.24, -.16 t -.3, and -.32 t -.85, respectively. Critical t the interpretatin f the results f the present study, the errr in the in-plane shear strain was always less than.1, and the errr in the in-plane angle ranged frm -3.9 t 3.4. As mentined abve, the errrs fr the cylindrical mdel were smewhat smaller and, perhaps, mre representative f the experimental errrs ccurring in the marker implantatin regin n the anterir free wall where the curvature is much smaller in the lngitudinal than in the circumferential directin. Thus, the errrs intrduced by the unifrmity assumptin wuld nt influence ur majr findings substantially. In the present study, fiber rientatin and threedimensinal finite defrmatin have been examined in the anterir wall f the canine left ventricle. The principal directin f defrmatin assciated with the greatest shrtening has been cmpared with the lcal fiber directin in the uter and inner halves f the ventricular wall. Althugh fiber directin and principal directin are nt substantially different in the uter half, they are virtually rthgnal t each ther in the inner half f the wall. Nevertheless, substantial shrtening defrmatin alng the fiber directin tends t ccur during cntractin with a magnitude abut half f its value alng the crssfiber directin. These data suggest that bth rerientatin (relative mtin) and crss-sectinal shape changes f mycytes r interstitium may cntribute t the large wall thickenings bserved during cntractin, particularly in the inner half f the ventricular wall. Acknwledgments We gratefully acknwledge the surgical and technical assistance f Richard Pavelec and Henry Mnre. We als thank Lana Nim and Dr. Eric Breisch fr help with histlgical techniques. References 1. Caulfield JB, Brg TK: The cllagen netwrk f the heart. Lab Invest 1979;4: Rbinsn RF, Winegard S: A variety f intercellular cnnectins in heart muscle. J Ml Cell Cardil 1981 ;13: Streeter DD: Grss mrphlgy and fiber gemetry f the heart, in Berne RM (ed): Handbk f Physilgy, Sectin 2: The Cardivascular System, Vlume I. Washingtn, DC, American Physilgical Sciety, 1979, pp Streeter DD, Sptnitz HM, Patel DJ, Rss J Jr. Snnenblick EH: Fiber rientatin in the canine left ventricle during diastle and systle. Circ Res 1969;24: Carew TE, Cvell JW: Fiber rientatin in the hypertrphied canine left ventricle. AmJPhysil 1979;236:H487-H Waldman LK, Cvell JW: The effects f ventricular pacing n finite defrmatin in the canine left ventricle. Am J Physil 1987;252:H123-H13 7. Waldman LK, Fung YC, Cvell JW: Transmural mycardial defrmatin in the canine left ventricle. Circ Res 1985; 57: Gallagher KP, Osakada G, Hess OM, Kzil JA, Kemper WS, Rss J Jr: Subepicardial segmental functin during crnary stensis and the rle f mycardial fiber rientatin. Circ Res 1982^: Freeman GL, LeWinter MM, Engler RL, Cvell JW: Relatinship between mycardial fiber directin and segment

13 562 Circulatin Research Vl 63, N 3, September 1988 shrtening in the midwall f the canine left ventricle. Circ Res 1985^6: Arts T, Veenstra PC, Reneman RS: Epicardial defrmatin and left ventricular wall mechanics during ejectin in the dg. Am J Physil 1982;243:H379-H39O 11. Lew WYW, LeWinterMM: Reginal cmparisn f midwall and area shrtening in the canine left ventricle. Circ Res 1986;58: Dieudnne JM: Gradients de directins et de defrmatins principales dans la pari ventriculaire gauche nrmale. J Physil (Paris) 1969;61: Fentn TR, Cherry JM, Klassen GA: Transmural mycardial defrmatin in the canine left ventricular wall. Am J Physil 1978;235:H523-H Meier GD, Ziskin MC, Santamre WP, Bve AA: Kinematics f the beating heart. IEEE Trans Bimed Eng 198; 27: Villareal F, Waldman L, Cvell J, Lew W: A new technique fr measuring reginal tw-dimensinal finite strains and strain directin in the canine left ventricle (abstract). Fed Amer Sc Exp Bi, April Yran C, Cvell JW, Rss J Jr: Rapid fixatin f the left ventricle: Cntinuus angigraphic and dynamic recrdings. JAppl Physil 1973;35: Stwell RE: Effect n tissue vlume f varius methds f fixatin, dehydratin, and embedding. Stain Technl 1941; 16: Ingels NB Jr, Daughters GT, Davies SR: Sterephtgrammetric studies n the dynamic gemetry f the canine left ventricular epicardium. J Bimech 1971 ;4: Hattri S, Weintraub WS, Aggarwal JB, Bdenhcimer MM, Banka VS, Helfant RH: Cntrasting ischemic cntractin patterns by zne and layer in the canine mycardium. Am J Physil 1982;243:H852-H Sptnitz HM, Sptnitz WD, Cttrell TS, Spir D, Snnenblick EH: Cellular basis fr vlume related wall thickness changes in the rat left ventricle. J Ml Cell Cardil 1974; 6: Hutchins GM, Mre GW, Hattn EV: Arterial-venus relatinships in the human left ventricular mycardium: Anatmic basis fr cuntercurrent regulatin f bld flw. Circulatin 1986;74: Prinzen FW, Arts T, Prinzen TT, Reneman RS: Cmments n "Relatinship between mycardial fiber directin and segment shrtening in the midwall f the canine left ventricle." Circ Res 1985;57: Prinzen FW, Arts T, Van der Vussc GJ, Reneman RS: Fiber shrtening in the inner layers f the left ventricular wall as assessed frm epicardial defrmatin during nrmxia and ischemia. J Bimech 1984,17: Arts T, Meerbaum S, Reneman RS, Crday E: Trsin f the left ventricle during ejectin phase in the intact dg. Cardivasc Res 1984;18: Wiseman MD, Hunter WC, Duglass AS: Errrs in calculatin f lcal mycardial strain derived frm the psitins f 4 implanted markers (abstract). Fed Amer Sc Exp Bi, Washingtn, DC, April 1987 KEY WORDS ventricular functin transmural defrmatin fiber rientatin