283:1609-1615, 2002. First published Jun 20, 2002;  doi:10.1152/ajpheart.00239.2002 AJP - HeartPrinzen, J. Maessen and T. Arts A. Van der Toorn, P. Barenbrug, G. Snoep, F. H. Van der Veen, T. Delhaas, F. W.in aortic valve stenosis patients using MRI tagging Transmural gradients of cardiac myofiber shortening

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Transmural gradients of cardiac myofiber shorteningin aortic valve stenosis patients using MRI tagging

A. VAN DER TOORN,1 P. BARENBRUG,3 G. SNOEP,4 F. H. VAN DER VEEN,3

T. DELHAAS,2 F. W. PRINZEN,2 J. MAESSEN,3 AND T. ARTS1

Cardiovascular Research Institute, Departments of 1Biophysics and 2Physiology, MaastrichtUniversity, 6200MD Maastricht; and Departments of 3Cardiothoracic Surgery and4Radiology, Academic Hospital Maastricht, 6202AZ Maastricht, The NetherlandsReceived 25 February 2002; accepted in final form 12 June 2002

Van der Toorn, A., P. Barenbrug, G. Snoep, F. H. Vander Veen, T. Delhaas, F. W. Prinzen, J. Maessen, and T.Arts. Transmural gradients of cardiac myofiber shorteningin aortic valve stenosis patients using MRI tagging. Am JPhysiol Heart Circ Physiol 283: H1609–H1615, 2002. Firstpublished June 20, 2002; 10.1152/ajpheart.00239.2002.—Aortic valve stenosis impairs subendocardial perfusion witha risk of irreversible subendocardial tissue damage. A likelyprecursor of damage is subendocardial contractile dysfunc-tion, expressed by the parameter TransDif, which is definedas epicardial minus endocardial myofiber shortening, nor-malized to the mean value. With the use of magnetic reso-nance tagging in two short-axis slices of the left ventricle(LV), TransDif was derived from LV torsion and contractionduring ejection. TransDif was determined in healthy volun-teers (control, n � 9) and in patients with aortic valvestenosis before (AVSten, n � 9) and 3 mo after valve replace-ment (AVRepl, n � 7). In the control group, TransDif was0.00 � 0.14 (mean � SD). In the AVSten group, TransDifincreased to 0.96 � 0.62, suggesting impairment of subendo-cardial myofiber shortening. In the AVRepl group, TransDifdecreased to 0.37 � 0.20 but was still elevated. In eight ofnine AVSten patients, the TransDif value was elevated indi-vidually (P � 0.001), suggesting that the noninvasively de-termined parameter TransDif may provide important infor-mation in planning of treatment of aortic valve stenosis.

aortic valve replacement; function; torsion

MOST CARDIAC DISORDERS affect motion of the left ventric-ular (LV) wall (18, 19, 25). In aortic valve stenosis,coronary flow diminishes in the subendocardial layersrelative to subepicardial ones (9, 16, 26). Ultimately,this may lead to subendocardial infarction and otherstructural changes (13–15). The risk to run into per-manent damage is the most important reason for aorticvalve replacement. Currently, this risk is estimated bycombined assessment of hemodynamic parameters(peak systolic LV pressure, transvalvular pressuredrop, and peak flow velocity), general LV pump func-tion parameters (ejection fraction and maximum timederivative of LV pressure), effects of increased venouspressure (edema), and sound from turbulent flow. In

the search for a clear indicative parameter, measure-ment of subendocardial contractile function may bepromising. After all, underperfusion of myocardial tis-sue results in an immediate decrease of contractilefunction (10, 22) far before irreversible tissue damageoccurs. The present study was conducted to investigatewhether measurement of subendocardial contractiledysfunction is possible and whether it may provideadditional clinically valuable information.

Commonly, myocardial contractile dysfunction is de-tected from wall motion abnormalities, as measured byechocardiography. In aortic valve stenosis patients,such abnormalities cannot be found easily, probablybecause the related contractile dysfunction is distrib-uted smoothly over the whole inner side of the LV wall.In the last decade, the MRI-tagging technique has beendeveloped to assess myocardial motion noninvasively(18, 20, 27). With this technique, during a period of afew milliseconds, the magnetization of the tissue isperiodically modulated with a grid pattern, thus form-ing tags. During the cardiac cycle, the tags are movingwith the tissue and are visualized with MRI. With theuse of image analysis techniques, the tags are followedin time. Thus a map of motion of the visualized crosssection is obtained. In the current study, we propose touse MRI tagging in detecting transmural differences inmyocardial contractile function.

In model studies on cardiac mechanics (3, 5), it waspredicted that a transmural gradient in contractilefunction was related to changes in the ratio of torsionto shortening (TSR) of the LV. Torsion is defined hereas a base-to-apex gradient of rotation of the short-axiscross section around the LV long axis. When assumingmyofiber shortening to be transmurally uniform, TSRwas predicted to be a fixed number. In dogs withtwo-dimensional echocardiography (3) and in humanswith MRI tagging (1), this prediction was confirmed. Inhumans using MRI tagging, the transmural relativevariation of myofiber shortening was reported to be�10% (17, 23). In aortic valve stenosis patients, torsionwas found to be elevated (21, 24). Torsion also changed

Address for reprint requests and other correspondence: T. Arts,Dept. of Biophysics, PO Box 616, 6200 MD Maastricht, The Nether-lands (E-mail: t.arts@bf.unimaas.nl).

The costs of publication of this article were defrayed in part by thepayment of page charges. The article must therefore be herebymarked ‘‘advertisement’’ in accordance with 18 U.S.C. Section 1734solely to indicate this fact.

Am J Physiol Heart Circ Physiol 283: H1609–H1615, 2002.First published June 20, 2002; 10.1152/ajpheart.00239.2002.

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by changes in preload, afterload, or contractility (8).We conclude that under normal conditions myofibershortening is quite evenly distributed over the LV wall,a finding that was confirmed experimentally by findinga fixed ratio of TSR. During aortic valve stenosis,torsion may be modified by changes in the transmuraldistribution of myofiber shortening. In the presentstudy, we investigated whether a parameter express-ing a transmural difference in myofiber shortening canbe derived from myocardial motion, as measured byMRI tagging.

In the normal heart, due to the specific helical path-ways of the myofibers, torsion as induced by the sub-epicardial myofibers is partly counteracted by the sub-endocardial myofibers. With aortic valve stenosis,subendocardial myofibers are likely to be less active,causing torsion to increase. Because fewer fibers arefully active, ejection is expected to decrease. Thus TSRis expected to be sensitive to a decrease of myofibershortening in the subendocardium relative to the sub-epicardium. If load of the heart changes by physiolog-ical mechanisms such as exercise or contractility vari-ation, myofiber shortening is expected to changeuniformly throughout the heart. Both torsion andshortening are then expected to change proportionally,leaving TSR unaffected.

In the present study, TSR was determined from themotion of MRI tags in two parallel short-axis crosssections. With the use of a model of LV mechanics, fromthe TSR value, the transmural difference in myofibershortening (TransDif), as normalized to the meanvalue, was estimated. Measurements were performedin healthy volunteers (control) and in aortic valve ste-nosis patients before (AVSten) and 3 mo after aorticvalve replacement (AVRepl). It was investigatedwhether TSR and TransDif were affected by aorticvalve stenosis. Finally, we investigated whether sub-endocardial myofiber shortening recovered after aorticvalve replacement.

METHODS

Patient study. The study was conducted on 9 healthyvolunteers (2 women and 7 men, age 31 � 4 yr) and 10patients (2 women and 8 men) with severe aortic valvestenosis, characterized by an aortic valve pressure gradient�50 mmHg. The AVSten group of patients (age 64 � 9 yr)contained nine patients before aortic valve replacement. TheAVRepl group of patients (age 68 � 6 yr) contained sevenpatients 3 mo after aortic valve replacement. From the lattergroup, six individuals also belonged to the AVSten group.Written informed consent was obtained from all subjects.

Image acquisition. MRI experiments were performed ateither 0.5 or 1.5 T (Philips Gyroscan T5 II or Gyroscan NT,Philips Medical Systems; Best, The Netherlands). All imageswere acquired using ECG triggering while patients werebreathing normally. Images of two short-axis cross sectionsof the heart were obtained, �2 and 4 cm below the base.Sixteen regular cine images per slice were acquired using anontagged steady-state gradient echo sequence, starting 20ms after the ECG trigger on the peak of the R wave andcontinuing with time intervals of either 20 or 62 ms (echotime 10 ms, slice thickness 8 mm, field of view 213 mm, image

size 256 � 256 pixels). Thereafter, a series of tagged imagesfrom the same two slices were obtained with time intervals of20 ms, using spatial modulation of the magnetization (6).Initial intertag distance was 5 mm with a tag width of 2.5mm in both horizontal and vertical directions. To obtain asufficient contrast-to-noise ratio for a longer time interval,two overlapping series of tagged images were acquired foreach slice. Acquisition of the first set of MRI-tagged imageswas started 20 ms after the ECG trigger. The second set ofMRI-tagged images started 130–160 ms later.

Image analysis. MR images were analyzed to quantifytorsion and the decline in cavity area, as normalized to wallarea. Image analysis was performed off-line using homemadesoftware for Matlab 5.3.1 (MathWorks; Natick, MA). For bothslices, a nontagged reference image was chosen at �160–180ms after the ECG trigger. In this image, the LV wall wasmanually outlined for both slices (Fig. 1). Papillary muscleswere considered not to be part of the LV wall. MRI imageswere recorded in the period from 30 ms until �260 ms afterthe ECG trigger with regular time intervals of 20 ms. Eachimage frame was split into an image with horizontal and animage with vertical tag lines by band filtering in the fre-quency domain around the first harmonic frequency of thetag pattern (spatial frequency 0.2 mm�1, ratio of bandwidthto center frequency 1.0). With the use of a correlation methodpreviously applied for pulsed ultrasonic echo signals (7), fromthe successive images with horizontal tag lines, vertical dis-placement was calculated in each pixel to obtain verticaldisplacement maps for each time interval. Similarly, theimages with vertical tags were used to obtain horizontaldisplacement maps. Because the boundaries of the LV wallmove with the tags, the calculated displacement maps wereused to determine the moving contours of the LV wall duringthe cardiac cycle. Thus the cross-sectional areas of the cavity(Acav) and of the LV wall (Awall) were determined as a func-tion of time for slice 1 and slice 2, which were located at thebasal and apical side of the equator, respectively. Shorteningwas quantified as the change (�) in the logarithm of thenormalized inner diameter (inner). It holds that

εinner � 1⁄2 ln �Acavslice 1 � Acav

slice 2

Awallslice 1 � Awall

slice 2� (1)

Time-dependent rigid body rotations of slice 1 and slice 2were also derived from the displacement maps. Torsion wasdefined as the shear angle on the epicardial surface betweenboth slices (3). The time course of torsion (T) was calculated

Fig. 1. MRI cine image (A) and MRI-tagged image (B) of a healthyvolunteer 178 ms after the peak R wave of the ECG. The wall, asdelineated manually in the cine image, was used for further motionanalysis.

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from the axial gradient in rotation angle multiplied by theouter radius (Repi) using the approximation

T ��slice 1 � slice 2�

d�

�Repislice 1 � Repi

slice 2�

2(2)

with

Repi � �Awall Acav

�

The parameter d represents the distance between both slices.By plotting torsion as a function of inner, a practically linearrelationship was found (1) during ejection. In determinationof the ejection phase, the beginning ejection usually started�40 ms after the ECG trigger. The end of ejection occurred�220 ms later, as marked by the end of decline of cavity area.The TSR was calculated as the negative slope of the linearregression line of torsion as a function of inner during theejection phase

T � �TSR � εinner intercept error (3)

The intercept has not been used for further analysis. Theerror represents mismatch, from which the sum of squareshas been minimized by the linear regression method.

Relation between TSR and TransDif. In the normal heart,myofiber strain is about uniformly distributed. Myofibers runapproximately circumferentially at midwall, which is definedhere by equal cross-sectional areas inside and outside themidwall radius (4). Mean natural myofiber strain (�f, mean)is expressed as a change (�) in

εf, mean � 1⁄2 ln �1 2Acav

Awall� (4)

The normalized TransDif in natural myofiber strain has beenquantified as

TransDif ��εf, epi � �εf, endo

�εf, mean(5)

where �f, epi and �f, endo refer to the epicardial and endo-cardial myofiber strain, respectively. A model of LV wallmechanics has been used to express TransDif as a function ofTSR and the ratio of cavity to wall area (Acav/Awall), resultingin the following approximation (APPENDIX)

TransDif � �1.57 � 1.16 �Acav

Awall�

me� �TSR

TSR0� 1� (6)

TSR0 represents the normal value of TSR, which appeared tobe 0.444 � 0.072 in the control group of the present study.The subscript me refers to midejection, which is the averagebetween the begin and end of ejection (APPENDIX).

Statistical analysis. Differences were analyzed for signifi-cance using a two-tailed t-test, supposing unequal variances.Within the group of patients, differences between before andafter aorta valve replacement were evaluated intraindividu-ally. Values are presented as means � SD.

RESULTS

Figure 2 shows representative results of analyzingthe MRI-tagged images for the LV of a healthy volun-teer. The top tracings show wall and cavity area for thetwo short-axis sections of the heart as a function oftime during the heart cycle. With the use of Eq. 1, fromthese areas the inner was determined. After �260 ms,inner reaches a plateau, indicating the end of the ejec-

tion phase by closure of the aortic valve. In Fig. 2, rigidbody rotations for both slices and calculated torsion(Eq. 2) are plotted as a function of time. In Fig. 3,torsion is plotted as a function of inner. The slope of thelinear regression line through these data points repre-sents the negative of the TSR value (Eq. 3).

In Fig. 4, torsion is plotted as a function of thelogarithm of inner for the control, AVSten, and AVReplgroups. The slopes of the linear regression lines reflectthe individual TSR values. In Table 1, the mean valuesand SDs for these groups are presented for age, Awall,Acav, inner, torsion, TSR, and TransDif. If applicable,values are presented for the beginning and end ofejection. Both beginning and end of ejection values ofinner in the control group were larger (P � 0.05) andvaried less than in the AVSten and AVRepl groups.Apparently, the LV of the patients was concentricallyhypertrophied with no significant difference betweenthe AVSten and AVRepl groups. In the control group,calculated myofiber shortening appears to be in a nar-row range (0.142 � 0.010). In both the AVSten andAVRepl groups, myofiber shortening is less than in thecontrol group, P � 0.0001 and P � 0.02, respectively.Myofiber shortening in the AVSten and AVRepl groupswas not different as a group as well as intraindividu-ally.

In the control group, TSR (0.444 � 0.073) was lowestwith the smallest variance. In the AVSten group, theTSR value (0.85 � 0.26) was elevated (P � 0.001) and

Fig. 2. Left ventricular (LV) motion parameters during ejection in anormal volunteer. From the short-axis motion data, wall area (opensymbols), cavity area (closed symbols), and rotation (open symbols)are determined as a function of time (top tracings). The squares andcircles refer to the short-axis slices 2 and 4 cm below the base,respectively. The logarithm of normalized inner diameter (inner) isone-half the logarithm of the cavity area as normalized to the wallarea. Torsion is the shear angle at the epicardium, which is propor-tional to the difference in rotation between both slices.

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variable. In the AVRepl group, the TSR value (0.61 �0.09) decreased (P � 0.02) but was still elevated rela-tive to the control group (P � 0.001).

When estimating TransDif according to Eq. 5 (Fig.5), this variable was 0.00 � 0.14 in the control group,0.96 � 0.62 in the AVSten group, and 0.37 � 0.20 in theAVRepl group. Statistical differences between thegroups were very similar to those of the parameterTSR. In all patients being monitored before and afteraortic valve replacement, TransDif decreased, indicat-ing development toward a more uniform transmuraldistribution of myofiber shortening.

DISCUSSION

In patients, the TSR of the LV during the ejectionphase was determined noninvasively using MRI tag-ging. In the presence of an aortic stenosis, TSR wassignificantly higher than in a group of healthy volun-teers. After aortic valve replacement, TSR partiallyreturned to normal. With the use of a model of cardiacmechanics (APPENDIX), the value of TSR was convertedto the parameter TransDif, which expresses the epicar-dial-endocardial difference in myofiber shortening asnormalized to mean myofiber shortening. In patientswith aortic valve stenosis, TransDif was elevated rel-ative to the control group, indicating severe contractiledysfunction of the inner layers. After aortic valve re-placement, TransDif decreased to an intermediatevalue, indicating an important recovery of contractilefunction of the inner layers. MRI tagging appears thefirst technique by which a transmural difference inmyofiber shortening can be quantified noninvasively inhumans.

Patients with aortic valve stenosis have elevatedend-diastolic concentricity of the LV (P � 0.01), imply-ing that the wall is thick relative to the cavity diame-ter. Concentricity is the opposite of dilatation, which isquantified by inner (Eq. 1; Table 1). Three months aftervalve replacement, end-diastolic concentricity waspractically unchanged and still elevated (P � 0.02).Thus, after aortic valve replacement, recovery of sub-

endocardial contractile function appears faster thanregression of concentricity. Besides concentricity, inthe AVSten group, the wall area was found to increase,indicating development of hypertrophy (P � 0.01),which partly recovered in the AVRepl group.

Because the biological variance of TransDif for nor-mals was so small, for eight of nine AVSten patientsthe individual TransDif value was significantly higherthan the value in the control group (P � 0.001), assum-ing a Gaussian distribution of the TransDif values inthe control group. After aortic valve replacement, inonly two of seven patients was this level of significancereached, whereas in the control group all individualvalues obviously were normal. Thus the presented MRItagging technique seems well suited to detect suben-docardial pathology in an individual patient.

No other parameter was found to be as conclusive asTransDif to distinguish individual AVSten patientsfrom the control group. TSR is close and was able todetect seven of nine patients with P � 0.001. In thecontrol group, calculated f, mean as well as torsion

Fig. 3. Torsion plotted as a function of inner for the data presented inFig. 2. The relation appears practically linear. The slope of the linearregression line represents the negative of the torsion-to-shorteningratio (TSR). Subendocardial dysfunction causes the TSR to increase.

Fig. 4. Torsion plotted as a function of inner for all volunteers andpatients. The control, aortic valve stenosis (AVSten), and aortic valvereplacement (AVRepl) groups are separated for convenience by avertical shift. The thick solid lines represent the average curves ofthe different groups. TSR values, as derived from the slopes, areminimum in the control group with little variation. In the AVStengroup, TSR is high and has a larger range. In the AVRepl group, TSRpartially recovers, as shown by a lower slope and decreased variabil-ity. In both patient groups, concentricity is elevated similarly, asshown by the shift of the curves to the left. The six different symbolsin the AVSten and AVRepl groups refer to the same patients. Onepatient (}) had also a left bundle branch block.

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(Table 1) appeared to be in a narrow range, 0.142 �0.010 and 0.114 � 0.034, respectively. In the AVStengroup, these values were changed highly significantly,P � 0.0001 and P � 0.002, respectively. Nevertheless,because of the large spread in the control group, theseparameters were less suited to detect individual pa-tients. Only four and one of nine patients were found tobe abnormal, respectively.

During ejection, measured motion of the LV wallcaused the cross-sectional area of the wall to decreaseslightly (Table 1). This decrease is likely due to anartifact in the measurement of radial displacement

near the inner and outer boundaries of the wall. Atthose locations, the MRI tags are often malformed,introducing errors in displacement calculation. Thisdisplacement has been used to follow the contours ofthe wall. The method of analysis has been set up suchthat the influence of this error is moderate and does notaffect any major conclusion of this study.

Control measurements were performed in young vol-unteers, whereas the patients were much older (Table1). Current insights suggest that the TSR is indepen-dent of age and species. Assuming homogeneity ofmechanical load in the heart, TSR was predicted in amathematical model of LV mechanics (3) without usingany species-specific information. TSR seems a mathe-matically determined constant that is expected to beuniversal to all hearts, “from mice to elephants.” Themathematically predicted value was found not to bedifferent from experimental measurements in normalhearts of dogs (3) and of young volunteers (1). It hasalso been reported that both torsion (12) and ejectionfraction in mice are not different from that in humans,suggesting similarity of TSR. Nevertheless, it is wise toperform measurements of TSR in an age-matchedgroup of humans. Unfortunately, these data are notavailable yet.

The presented MRI-tagging technique is the firstclinically feasible technique to quantify transmuraldifferences in myocardial contraction. The MRI proto-col takes �25 min, finding the position and orientationof the heart inclusive. In the current setup, time isneeded for selection and transfer of files. After imageacquisition, epi- and endocardial contours have to bedrawn manually in one set of images, which takes �5min. This step is the most cumbersome one and should

Table 1. Summary of measurements

Control (n � 9) AVSten (n � 9) AVRepl (n � 7)

Mean �SD Mean �SD Mean �SD

Age, yr 31 4 64 9 68 6Awall, mm2

Beginning of ejection 2,302 372 2,859 472 2,578 360End of ejection 2,289 391 2,755 493 2,428 298

Acav, mm2

Beginning of ejection 1,843 327 1,621 366 1,576 425End of ejection 1,100 237 1,033 265 941 404

εinner

Beginning of ejection �0.107 0.084 �0.282* 0.162 �0.260* 0.318End of ejection �0.365 0.120 �0.507* 0.175 �0.517* 0.471

εf, mean

Beginning of ejection 0.135 0.050 0.039 0.085 0.051 0.072End of ejection �0.006 0.054 �0.067 0.074 �0.067 0.087Difference 0.142 0.0104 0.107† 0.022 0.117* 0.018

Torsion, radBeginning of ejection 0.004 0.007 0.016 0.008 0.006 0.004End of ejection 0.118 0.033 0.200 0.053 0.165 0.050Difference 0.114 0.034 0.184† 0.054 0.159* 0.050

TSR 0.444 0.073 0.853† 0.264 0.605† 0.085TransDif 0 0.14 0.96† 0.62 0.37† 0.20

n � No. of healthy volunteers (control) and patients with aortic value stenosis before (AVSten) and 3 mo after aortic value replacement(AVRepl). Awall, wall area; Acav, cavity area; εinner, logarithm of normalized inner diameter; εf, mean, mean natural myofiber strain; TSR,torsion-shortening ratio; TransDif, normalized transmural differences of fiber shortening. *P � 0.05 and †P � 0.001, significantly differentfrom the control group.

Fig. 5. Normalized transmural difference in myofiber shortening(TransDif; Eq. 5) in the control, AVSten, and AVRepl groups. Theopen circles and error bars indicate means and SDs. Lines connectcorresponding patient data.

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be automated for easy clinical application. Finally,image and data analysis take another 5 min on aregular personal computer. Various parts of the proto-col and analysis may be improved, such as the use ofbreath-hold in MRI tagging.

In conclusion, using MRI tagging, the motion of twocross sections of the LV was measured noninvasively inhealthy volunteers (control) and in AVSten andAVRepl patients. TSR in the AVSten group was signif-icantly larger than in the control group. In the AVReplgroup, TSR declined but did not return completely.With the use of a model of LV mechanics, from TSRand Acav/Awall, TransDif was calculated as normalizedto mean myofiber shortening. In the control, AVSten,and AVRepl groups, TransDif was 0 � 0.14, 0.96 �0.62, and 0.38 � 0.20, respectively. Thus, in aorticvalve stenosis, decreased subendocardial contractilefunction was found to recover 3 mo after aortic valvereplacement. Over the same period, recovery of theshape of the heart (concentricity) appeared insignifi-cant. In the AVSten group, eight of nine patients wererecognized by a significant reduction of TransDif (P �0.001) relative to the control group.

APPENDIX

Relation between TSR and TransDif. The value of the TSRwas derived directly from the MRI tagging data. In thisappendix, TSR will be related to the transmural difference inmyofiber shortening as expressed by the parameter TransDif(Eq. 4). Consider the equatorial wall as a thick-walled cylin-der section with inner radius ri and outer radius ro. Wallcoordinates [r, c, z] represent the radial, circumferential, andbase-to-apex directions. For small rotational symmetric de-formations of an incompressible cylinder, circumferentialstrain cc, base-to-apex strain zz, and shear angle cz can beexpressed as a function of radius r as

εcc(r) � �A Br�2 εzz � 2A εzc(r) � Cr (7)

Parameters A, B, and C are constants. To solve the problem,a sufficient number of equations has to be found. For smalldeformations, myofiber strain (f) depends on a change in theratio of cavity volume (Vcav) to wall volume (Vwall) (2). Withthe use of Eq. 7, and by substituting the inner and outerradius of the cylinder section, it is found that

εf, mean ��(Vcav/Vwall)

1 3Vcav/Vwall�

�(Acav/Awall)1 3Acav/Awall

�2B

ro2 � 2ri

2 (8)

Given a myofiber angle , for myofiber strain it holds that

εf (r, ) � εcc cos2 εzz sin2 εzc sin cos (9)

For the normal heart, f is assumed to be the same every-where (4). Thus three equations were obtained for the innersurface (r � ri), for the midwall surface (r � rmid), with 2rmid

2 �(ri

2 ro2), and on the outer surface (r � ro)

εf (ri, i) � εf, mean εf (rmid, 0) � εf, mean

εf (ro, o) � εf, mean

(10)

At midwall, the myofibers are directed circumferentially (11),implying that mid � 0. The parameter TSR is the ratio ofshear at the outer surface and relative inner diameter de-crease

TSR �εzc (ro)εcc (ri)

(11)

Now assume that ri, ro, and TSR0 are known. Equations 8, 10,and 11 then provide five equations with five unknowns, i.e.,A, B, C, i, and o. This system of equations was solved usingMathematica 4.1 (Wolfram Research; Champaign, IL). As aresult, the fiber angles i and o are known.

In the control group for midejection, the ratio of cylindercavity to cylinder wall volume was (Acav/Awall)me � 0.64 andTSR0 � 0.444 (Table 1). With the use of these data, it wasderived that i � 0.68 rad and o � �0.94 rad. Given adifferent value of TSR, the value of C in Eq. 7 changes,causing only shear zc to change. With Eq. 9, myofiber strainwas calculated numerically at the inner and outer surfaces.With Eq. 5, the parameter TransDif was calculated. For theoccurring range (Table 1), the following approximation wasfound

TransDif

� �1.20 � 1.16 �Acav

Awall�

me

0.82TSR0� �TSRTSR0

� 1� (12)

By substituting TSR0 � 0.444, this equation results in Eq. 6.

This work was supported by The Netherlands Heart FoundationGrant 98.035.

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