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Am J Physiol Heart Circ Physiol
280: H2639–H2648, 2001.
Compression of interventricular septum during
right ventricular pressure loading
GREGORY S. NELSON,1 EZZELDIN Y. SAYED-AHMED,2 CAROL A. GIBBONS KROEKER,1
YI-HUI SUN,1 HENK E. D. J. TER KEURS,1 NIGEL G. SHRIVE,1 AND JOHN V. TYBERG1
1
Departments of Medicine, Physiology and Biophysics and Civil Engineering, University of
Calgary, Calgary, Alberta T2N 4N1, Canada; and the 2Department of Civil Engineering,
Ain Shams University, Cairo 11331, Egypt
Received 19 July 2000; accepted in final form 24 January 2001
(17) suggested in 1914 that displacement of the septum during diastole might explain
how the filling of one ventricle influenced the performance of the other. The development of echocardiography led to renewed interest in septal mechanics, as
Popp et al. (33) observed, in patients with right ventricular (RV) volume overload (RVVO), that the septum
moved leftward during diastole, findings supported
later by others (4, 11, 14, 23, 37). Observations consistent with those described during RVVO also occurred
during RV pressure overload (RVPO). Stool et al. (40)
constricted pulmonary arteries in dogs and observed a
gradual increase in RV end-diastolic pressure and a
decrease in left ventricular (LV) septum-to-free wall
diameter, inferred to be the result of a leftward shift of
the septum. Other studies (3, 30) later reported similar
findings.
To define the determinants of septal position and
motion better, Kingma et al. developed an open-chest
canine model and investigated the effects of both
RVVO and RVPO on the position and motion of the
septum, as determined by sonomicrometry and twodimensional (2-D) echocardiography (24). Both RVVO
and RVPO progressively increased end-diastolic RV
septum-to-free wall diameter and complementarily decreased the respective LV diameter, in agreement with
previous studies (3, 4, 14, 30, 37, 40, 43). Furthermore,
they showed that the end-diastolic septal position (as
reflected by the septum-to-free wall diameters) was
precisely proportional to transseptal pressure (TSP ⫽
LVP-RVP). Normally, end-diastolic TSP is ⬃3 mmHg
positive and the septum is concave to the LV. However,
RVVO and RVPO reduced or reversed end-diastolic
TSP and shifted the septum leftward, flattening and
sometimes inverting it. Visner et al. (47) and Feneley
et al. (15) confirmed those results in their studies of
conscious dogs as well as Thompson et al.’s (45) study
of human patients. These studies seemed to suggest
that the septum behaved like a passive compliant
membrane (e.g., a sail) between two liquid-filled chambers in that its diastolic position could be predicted by
TSP.
Dong et al. (13) related septal short-axis radius of
curvature (R; 2-D echo) and midwall septal segment
length (SSL; sonomicrometry) to LVP, RVP, and ventricular septum-to-free wall diameter (2-D echo) during
pulmonary artery (PA) and aortic constrictions in anesthetized dogs. Over a wide range of TSP (0 ⫾ 7 mmHg),
they demonstrated that changes in ventricular diameters, as measured by Kingma et al. (24), were proportional to changes in SSL, which were also linear functions of TSP. Thus echo-measured changes in
Address for reprint requests and other correspondence: J. V. Tyberg, Depts. of Medicine and Physiology and Biophysics, Health
Sciences Centre, 3330 Hospital Dr. NW, Calgary, Alberta T2N 4N1
Canada (E-mail: [email protected]).
The costs of publication of this article were defrayed in part by the
payment of page charges. The article must therefore be hereby
marked ‘‘advertisement’’ in accordance with 18 U.S.C. Section 1734
solely to indicate this fact.
finite element analysis; interventricular septum; mechanics;
compressive stress
HENDERSON AND PRINCE
http://www.ajpheart.org
0363-6135/01 $5.00 Copyright © 2001 the American Physiological Society
H2639
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Nelson, Gregory S., Ezzeldin Y. Sayed-Ahmed, Carol
A. Gibbons Kroeker, Yi-Hui Sun, Henk E. D. J. Ter
Keurs, Nigel G. Shrive, and John V. Tyberg. Compression of interventricular septum during right ventricular pressure loading. Am J Physiol Heart Circ Physiol 280:
H2639–H2648, 2001.—The interventricular septum, which
flattens and inverts in conditions such as pulmonary hypertension, is considered by many to be an unstressed membrane, in that its position is assumed to be determined solely
by the transseptal pressure gradient. A two-dimensional
finite element model was developed to investigate whether
compression and bending moments (behavior incompatible
with a membrane) exist in the septum during diastole under
abnormal loading, i.e., pulmonary artery (PA) constriction.
Hemodynamic and echocardiographic data were obtained in
six open-chest anesthetized dogs. For both control and PA
constriction, the measured left ventricular and right ventricular pressures were applied to a residually stressed mesh.
Adjustments were made to the stiffness and end-bending
moments until the deformed and loaded residually stressed
mesh matched the observed configuration of the septum.
During PA constriction, end-bending moments were required
to obtain satisfactory matches but not during control. Furthermore, substantial circumferential compressive stresses
developed during PA constriction. Such stresses might impede septal blood flow and provoke the unexplained ischemia
observed in some conditions characterized by abnormal septal motion.
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SEPTAL COMPRESSION DURING RV LOADING
METHODS
Experimental Preparation
Six mongrel dogs of either sex were anesthetized, initially
with thiopental sodium (25 mg/kg iv), and subsequently with
an infusion of fentanyl citrate (25 mg 䡠 kg⫺1 䡠 h⫺1), intubated,
and ventilated (70% NO2-30% O2 mixture) with the use of a
constant-volume respirator (model 607, Harvard Apparatus;
Millis, MA). A large bore cannula was introduced into the
external jugular vein for administration of fluids. A midline
sternotomy was performed, and the ventrolateral surface of
the pericardium was incised transversely (⬃5-cm long) along
the base of the heart and retracted for instrumentation.
To manipulate TSP and achieve septal flattening and
inversion, an inflatable silicone occluder (In Vivo Metric;
Healdsburg, CA) was placed around the main PA. The heart
was then repositioned into the pericardium, and the pericardial margins were reapproximated with several sutures.
Care was taken to avoid decreasing the pericardial volume
(39).
LV and RV pressures were measured with 8-Fr micromanometer-tipped catheters with reference lumina (model
SPC-485, Millar Instruments; Houston, TX) inserted through
a carotid artery and internal jugular vein, respectively. A
liquid-filled catheter was inserted into the aorta via a femoral
artery to monitor pressure. The liquid-filled catheters were
attached to transducers (model P23ID, Statham Gould; Oxnard, CA) and referenced to the level of the right atrium.
Before the data were collected, pressures measured by the
micromanometers were matched to those measured from the
liquid-filled catheters to correct for baseline drift. After the
animals underwent surgical preparation, they were allowed
to stabilize for 30 min. Body temperature was maintained
throughout the experiment with the use of a heating pad, and
an electrocardiogram (lead II) was recorded to monitor cardiac rhythm and rate. All of the conditioned hemodynamic
signals were amplified (model VR16, Electronics for Medicine/Honeywell; White Plains, NY), passed through a lowpass filter (100 Hz), and digitized at 200 Hz.
Echocardiographic Frame Acquisition
The interventricular septum was imaged (30 frames/s)
echocardiographically (model V3400, Diasonics; Salt Lake
City, UT) in an LV minor-axis view by using a 5.0-MHz
transesophageal probe held on the surface of the RV free
wall. We were careful to maintain consistency during recording of the ultrasound images. Referencing to the level of the
tips of the papillary muscles (these served as “internal landmarks”) helped ensure that the same short-axis slice was
imaged in each of the six dogs. Furthermore, to account for
the effect of ventricular torsion during the cardiac cycle, the
imaging probe was rotated slightly, so that the same degree
of anterior and posterior septal insertion points was viewed
in each recording run.
A video frame synchronizer (Odessa Computer Systems;
Calgary, Canada) was used to match the echocardiographic
frames (recorded on 1⁄2-in. videotape) to be analyzed with the
corresponding hemodynamic data.
Experimental Protocol
Hemodynamic and echocardiographic data were obtained
simultaneously and continuously for 30 s (including a 10-s
control period) with the respirator turned off. Attempts were
made to obtain complete flattening or inversion of the septum
in each PA constriction run. Between interventions, an in-
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ventricular diameters reflect changes in the length of
the septum. Second, by plotting curvature (1/R) versus
TSP, they confirmed that the septum becomes flat only
when TSP approximates ⫺5 mmHg, consistent with
the results of other studies (1, 25).
Beyar et al. (6) demonstrated the importance of the
fact that the septum is a thick-walled structure, specifically that bending moments are important. Only
when their model accounted for bending moments did
it duplicate two critical features of the experimental
results: 1) that the septum is not flat at TSP ⫽ 0
mmHg, and 2) that midwall SSL continues to become
shorter after septal inversion, whereas end-diastolic
TSP becomes progressively negative. Although those
authors had expected reextension of SSL at greater
negative values of TSP, the observed result was consistent with the midwall SSL because it was subjected
to compressive stress. If SSL had been measured near
the LV endocardium, it would have relengthened, consistent with the extension stress borne by the LV
endocardial fibers during inversion. Thus, in contrast
to the implication of the previous echocardiographic
studies (24, 25, 43, 45), in which the septum behaved
like a passive compliant membrane, the septum
seemed to behave more like a thick-walled structure
subject to lateral pressure and end-bending moments.
That stress patterns are highly dependent on local
ventricular wall curvature was suggested by Gould et
al. (16). Later, Heethaar et al. (20) used a finite element (FE) model to study the stress gradients in both
single LV and in combined RV-LV models. The myocardium was assumed to be isotropic, homogeneous,
and linearly elastic. During diastole, the LV had the
largest stresses at the endocardial surface and, in the
combined RV-LV model, there was a large stress concentration at the septum-RV junction. Compressive
stresses were also found in the septum. Pao et al. (32),
also by using the FE approach, modeled a cross section
of an isolated LV in diastole (reconstructed from roentgenographic recordings) and assumed that the myocardium was homogeneous, isotropic, and linearly elastic.
Endocardium-to-epicardium circumferential stress
gradients occurred only in the anterior and posterior
walls. In the septum and LV free wall, the gradient
was reversed but of smaller magnitude. Furthermore,
the model showed that some circumferential compressive stresses developed near the LV septal surface,
where the wall curvatures were small and convex inward.
As described above, the motion pattern of the interventricular septum during pathophysiological (abnormal) loading conditions is complex. No study has thoroughly investigated this unique structure under such
severe conditions from a structural mechanics viewpoint. Accordingly, the goal of our study was to further
elucidate the structural behavior patterns of the septum under both normal and abnormal loading. Specifically, we developed a two-dimensional (2-D) FE model
of the interventricular septum to test the hypothesis
that secondary to PA constriction, compressive stress
and bending moments do develop in the septum.
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terval of 10–15 min was allowed to elapse for hemodynamic
stabilization.
Hemodynamic and Echocardiographic Data Analysis
Hemodynamic data were subsequently analyzed with the
use of commercial software (CVSOFT, Odessa Computer
Systems). With the use of image-analysis software developed
in our laboratory (CVIMAGE5, Odessa Computer Systems),
septal midwall points were identified on the 2-D echocardiograms. A midwall line was constructed by connecting these
points and by smoothing (2). From this line, we made radiusof-curvature measurements. Finally, thickness (normal to
the midwall line) was measured.
FE Model and Analysis
Fig. 1. A: finite element (FE) mesh used to model the interventricular septum composed of 120 plane-stress elements arranged in 6
layers of 20 elements. Truss elements (thick line) were used at both
ends of the mesh (left side, posterior insertion point; right side,
anterior insertion point) to allow for application of end-bending
moments. These truss elements were made sufficiently stiff so that
rigid-body rotation about the midnodes at the ends of the mesh could
be achieved. B: two different constitutive relationships were investigated: neo-Hookean (linear stress-strain) and sarcomere forcelength (nonlinear stress-strain) relationship (10), implemented via
an Arruda-Boyce energy function.
Bending Moment ⫽
冉
冊
␴RV ⫺ ␴LV T 2
2
6
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Model. A 2-D FE model of the interventricular septum was
developed, with the use of the FE program Abaqus (version
5.6, Hibbitt, Karlsson and Sorenson; Pawtucket, RI), to investigate the difference in structural behavior between control and PA constriction-loading conditions. The model will
be described only briefly here because a more detailed description has been published (29).
FE mesh constitutive relations. The FE mesh consisted of
120-plane stress elements arranged in 6 layers of 20 elements (Fig. 1A). Truss elements were used at both ends of the
mesh to allow for application of end-bending moments. These
truss elements were made sufficiently stiff so that rigid-body
rotation about the midnodes at the ends of the mesh could be
achieved. Two different constitutive relationships (Fig. 1B)
were investigated: 1) neo-Hookean (linear stress-strain) and
2) a sarcomere force-length (nonlinear stress-strain) relationship (10), implemented via an Arruda-Boyce energy function.
Both models were made hyperelastic to account in part for
any large strains associated with the PA constriction cycle.
Large deformation was also accounted for by invoking geometric nonlinearity into the analysis. In so-called linear
geometric problems, equilibrium equations may be written
with respect to the original undeformed configuration because the deformations are minimal (i.e., the configuration
remains essentially unchanged) (8). However, when the deformations become large, as occurs during abnormal loading
(i.e., PA constriction), equilibrium equations must be written
with respect to the deformed geometry.
Residual stress. Earlier investigators (31) have ignored the
residually stressed (RS) state of completely relaxed in situ
ventricular myocardium. We accounted for residual stress in
each dog by applying bending moments at the ends of the
initial unstressed mesh such that the resulting stress distribution closely matched that calculated by Guccione et al.
(18). The parameters of the calculated RS state were saved to
an input file and incorporated into each loading step in the
form of an initial stress condition.
Accounting for sarcomere length changes during diastole.
To analyze images other than those at end diastole, the
geometry of the RS mesh was adjusted according to values of
sarcomere length (SL) idealized from the measurements of
Rodriguez et al. (35) and Guccione et al. (19). In the cardiac
cycle, SL increases during diastole. Hence, the geometry of
the mesh had to be adjusted to account for diastolic SL
changes. On the basis of these studies (19), it was estimated
that SL increased 17% from its end-systolic value to its
end-diastolic value (15% linearly to the point of minimum LV
pressure when the LV fills rapidly and, perhaps, actively,
and a further 2% linearly to the point of minimum LV
pressure when the LV fills less rapidly and passively. Thus,
for each frame analyzed, the radius-of-curvature of the enddiastolic RS mesh was reduced accordingly. Furthermore,
because we assumed the septum maintains constant crosssectional area in the plane of the model, each radius-ofcurvature reduction coincided with a proportional increase in
thickness.
Loading conditions. Measured LV and RV pressures were
applied to the RS mesh and then end-bending moments were
adjusted until conformity to the observed configuration of the
septum was judged to be satisfactory. For both the neoHookean (linear) and nonlinear models, adjustments in material stiffness were also required for satisfactory matches to
be obtained.
Computational requirements: linear versus nonlinear
model. The approximate time required to reach convergence
for the neo-Hookean (linear) model was 15 min, whereas for
the sarcomere force-length (nonlinear) model, 2 h were required with the use of an Origin 2000 computer (Silicon
Graphics; Mountain View, CA). Because the results obtained
from both models were so similar (see Stress Plots and
Bending-Moment Diagrams), we accepted results with the
use of the neo-Hookean relationship for the multiple analyses
for each of the six dogs analyzed.
Bending-moment diagrams. Bending-moment diagrams
were constructed by calculating the bending moment per unit
thickness along the length of the septum by using the expression
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SEPTAL COMPRESSION DURING RV LOADING
where ␴RV is the circumferential stress at the RV septal
surface, ␴LV is the circumferential stress at the LV septal
surface, and T is the thickness.
Statistical Analysis
Hemodynamic and geometric data (presented as means ⫾
SE unless stated otherwise) were plotted with the use of
SigmaPlot 4.0 for Windows (Jandel Scientific; San Rafael,
CA) as a function of time to illustrate the temporal trends of
these parameters during control and PA constriction observations. All statistical testing was performed with the use of
SigmaStat 2.0 for Windows statistical software (Jandel) on a
Pentium IBM-compatible computer.
RESULTS
TSP and Septal Geometry
Fig. 2. Diastolic transseptal pressure gradient (TSP ⫽ LVP ⫺ RVP)
plotted as a function of time from end systole (ES) to end diastole
(ED) (pooled data from all 6 dogs). Compared with control data, TSP
during pulmonary artery (PA) constriction was substantially lower
(and negative) throughout diastole. In particular, TSP was reduced
near 0 mmHg at ES, dropped to a maximum negative value (approximately ⫺10 mmHg) shortly thereafter, approached a minimum
negative value near mid-diastasis, and then tended to become
slightly more negative toward ED.
Fig. 3. Curvature (1/R) of the midportion of the septum plotted
against time. During the control period (pooled data from all 6 dogs),
the curvature of the septum decreased from ES to ED. During PA
constriction (data from each dog plotted individually), the curvature
of the septum was reduced (i.e., the septum was flattened) compared
with the control data. Inversion was observed in only 3 of the 6 dogs.
In those dogs, inversion was first observed midway through diastole
(i.e., near mid-diastasis).
control data. Although TSP was lowered to a similar
degree during PA constriction in all dogs, inversion
was observed in only three dogs. In those dogs, inversion was first observed midway through diastole (i.e.,
near mid-diastasis).
Shear Modulus and End-Bending Moments
Once the measured loading was applied (LVP and
RVP) and changes in SL were accounted for (via
changes in mesh geometry), adjustments in shear modulus (G) (material stiffness for neo-Hookean formulation) and end-bending moments were required to make
the RS mesh conform satisfactorily to the observed
shape of the septum. In Fig. 4, G is plotted as a
function of time for both control and PA constriction
Fig. 4. Neo-Hookean shear modulus, G, plotted as a function of time
for both control and PA constriction cycles during diastole. For both
conditions, G decreased rapidly after ES, reached a minimum value
midway through diastole and then tended to increase slightly toward
ED.
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The diastolic (end systole to end diastole) transseptal
pressure gradient (TSP ⫽ LVP-RVP) is plotted in Fig.
2 (pooled data from all six dogs). Compared with control data, TSP during PA constriction was substantially lower (and negative) throughout diastole. In particular, TSP was reduced near 0 mmHg at end systole,
dropped to a maximum negative value (approximately
⫺10 mmHg) shortly after, approached a minimum negative value near middiastasis, and then tended to become slightly more negative toward end diastole.
Curvature (1/R) of the midportion of the septum is
plotted against time in Fig. 3. During the control period
(pooled data from all six dogs), the curvature of the
septum decreased from end systole to end diastole.
During PA constriction (data from each dog plotted
individually), the curvature of the septum was reduced
(i.e., the septum was flattened) compared with the
SEPTAL COMPRESSION DURING RV LOADING
H2643
cycles during diastole. In both loading conditions, G
decreased rapidly after end systole, reached a minimum value midway through diastole and then tended
to increase slightly toward end diastole.
End-bending moments are plotted as a function of
time in Fig. 5. Only the PA constriction data are presented, because no end-bending moments were required to make the control data RS meshes conform.
For all six dogs, greater end-bending moments (at both
insertion points) were required near end systole, compared with those required for the rest of diastole.
Larger end-bending moments were required at the
anterior insertion point (i.e., right side of the mesh, see
Fig. 1A) than at the posterior insertion point (i.e., left
side of the mesh).
To illustrate the difference in septal structural behavior between control and PA constriction loading,
representative principal-stress vector plots and corresponding bending-moment diagrams at mid-diastasis
are presented in Fig. 6. To minimize end effects, the
two columns of elements on either side of each mesh
are not shown. All of the arrows are equally scaled, the
magnitude being represented by the size of the paired
arrowheads and the distance between them. Whether
the stress is tensile or compressive is represented by
the direction of the arrowheads, where divergent
means tensile and convergent means compressive.
Both during control (see Fig. 6A; linear model) and
PA constriction observations (see Fig. 6, B and C;
linear and nonlinear model, respectively), the radial
compressive stresses that developed at the LV and RV
surfaces were equal to the applied LV and RV pressures (this was true for all dogs, in each loading condition, and at every point in diastole). The circumferential stresses, which developed in the control state,
were small in magnitude (they were similar to the
Fig. 6. To illustrate the difference in septal structural behavior
between control and PA-constriction loading, representative principal-stress vector plots (A, B, and C, top) and corresponding bendingmoment diagrams (A, B, and C, bottom) at middiastasis are presented. To minimize end effects, the 2 columns of elements on either
side of each mesh are not shown. All arrows are equally scaled, the
magnitude being represented by the size of the paired arrowheads
and the distance between them. Whether the stress is tensile or
compressive is represented by the direction of the arrowheads, divergent meaning tensile, convergent meaning compressive. See text
for further details.
Fig. 5. End-bending moments plotted as a function of time. Only the
PA constriction data are presented, because no end-bending moments were required to make the control-data RS meshes conform.
For all 6 dogs, greater end-bending moments (at both insertion
points) were required near ES, compared with those required for the
rest of diastole. Larger end-bending moments were required at the
anterior insertion point than at the posterior insertion point.
residual stresses) relative to PA constriction, where
much larger circumferential compressive stresses
clearly developed. These compressive stresses became
less circumferentially oriented in more lateral elements. In these elements, the compressive stresses
appeared to be aligned towards the “supports” of the
mesh (i.e., the midnodes of each end boundary). Note
how the stress distributions for the linear and nonlinear PA constriction models (see Fig. 6, B and C) are
practically identical.
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Stress Plots and Bending-Moment Diagrams
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SEPTAL COMPRESSION DURING RV LOADING
DISCUSSION
Compared with the stresses that developed during
the control cycle, the substantial circumferential compressive stresses, which developed in the PA constriction-loading conditions (Fig. 6), suggests an archlike
behavior of the septum under these severe conditions.
Recall that the arch is a structural system in which the
primary internal force is axial compression (7). Like an
arch, the septum bears the PA constriction loading by
distributing the net “downward” thrust of the load
laterally, through a pattern of compressive stress towards its “supports,” which, although not shown, seem
to be the RV insertions. Pao et al. (32) and Heethaar et
al. (20) have suggested compression in the septum.
However, the archlike distribution of stresses shown
here, with specifically, maximum compression occurring at the RV septal surface, is a novel finding. Furthermore, we have found with the model that this
behavior is sustained during diastole in all dogs analyzed, whereas many previous investigators have limited their analyses to end diastole only.
A degree of circumferential tension developed on,
and slightly below, the LV septal surface during PA
constriction when flattening and/or inversion was observed. This observation is consistent with that made
in a previous study from our laboratory, in which
Fig. 7. To represent the stress and strain data quantitatively, circumferential stress (A) and corresponding strain (B) values at middiastasis (pooled data from all 6 dogs) are plotted. Values were
obtained by averaging the principal stresses and strains at the
centroid of each of the 2 midlayer elements for all 6 layers across the
wall. During control, the circumferential stresses that developed
were roughly equivalent to the residual stress distribution (thick
line). Corresponding strains, which developed during control, were
small (i.e., all ⬍5%). For PA constriction, markedly different stress
and strain patterns were observed. Circumferential stress distribution, which developed was much larger and oppositely oriented,
relative to control.
septal segment length (SSL) was measured with the
use of sonomicrometry crystals that were positioned
circumferentially, approximately halfway between the
RV and LV septal surfaces (12). It was a surprise to
find that SSL did not lengthen, even with extreme
degrees of septal inversion. It had been assumed that
the septum behaved like a thin-walled membrane and
it therefore was reasoned that large negative values of
TSP would relengthen the segment, but this was never
observed. However, as was speculated at the time,
because the crystals had not been placed very close to
the LV septal surface, lengthening perhaps should not
have been expected.
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The bending-moment diagram for the control example revealed only slightly positive values for nearly
every element along the mesh (see Fig. 6A). In contrast, significantly different bending-moment distributions were observed for both PA constriction examples
(Fig. 6, B and C). Bending moments at the ends of the
meshes were positive and much larger than during
control observations, they crossed zero near elementpositions 5 and 14, and the most-negative bending
moment was achieved near the midpoint of the mesh.
The linear and nonlinear PA constriction models
showed practically identical bending-moment diagrams as the stress results did.
Finally, to represent the stress and strain data quantitatively for the linear model, transseptal circumferential stress and corresponding strain plots at middiastasis are shown in Fig. 7 (pooled data from all six
dogs). The values were obtained by averaging the principal stress at the centroid of each of the two midlayer
elements for all six layers across the wall. During
control, the circumferential stresses that developed
were roughly equivalent to the residual stress distribution (they were virtually superimposable), both
showing an approximately ⫺3 to ⫹3 mmHg stress
variation, from LV to RV septal surfaces, respectively.
The corresponding strains that developed during control were small (i.e., all ⬍5%). For PA constriction,
markedly different stress and strain patterns were
observed. The circumferential stress distribution that
developed was much larger and oppositely oriented
relative to control, showing an LV to RV septal surface
range of ⫹10 to ⫺28 mmHg and corresponding strains
of ⫹9 to ⫺17%.
SEPTAL COMPRESSION DURING RV LOADING
ture rather than the predominant circumferential
stress. For the control example, the circumferential
stress gradient from the LV septal surface to the RV
septal surface was compressive to tensile. Therefore,
the bending moment was calculated as positive. Consequently, the bending moment calculated for both
inverted septum examples varied from positive to negative to positive because the circumferential stress
gradient reversed at two locations. For element positions 3 and 4, the bending moment was positive (consistent with all end-bending moments at the posterior
insertion point, see Fig. 5) because the stress gradient
across the septum was compressive-to-tensile. From
element positions 5-14, the bending moment was negative because the stress gradient reversed, becoming
tensile-to-compressive. Finally, for element positions
15–18, the bending moment became positive again
(consistent with all end-bending moments at the anterior insertion point, Fig. 5) because the stress gradient
again became compressive-to-tensile. These results
make sense from a structural mechanics viewpoint. If
one considers a beam with fixed ends, subject to uniformly distributed loading (Fig. 8A) the deformation
and corresponding bending moment diagram (Fig. 8B)
are similar to those obtained by the model. Specifically,
flattening or inversion is greatest near the middle of
the septum, the zero-moment points coincide with the
points of inflection, and the maximum (absolute) bending moment is achieved at the ends (i.e., at the RV
insertions). This analogy suggests that the model is
predicting the structural behavior correctly, but perhaps more importantly, also confirms the necessity and
importance of including bending effects in the model.
We speculate that the substantial circumferential
compressive stresses predicted by the model during PA
constriction may pertain to a number of clinical situations. In several diseases, in which the TSP is decreased or reversed (e.g., mitral stenosis, primary pulmonary hypertension, and pulmonary thromboembolic
disease), and septal flattening and inversion is observed, there may be chest pain that is indistinguish-
Fig. 8. A: beam with fixed ends is subject to uniformly distributed
loading. B: deformation and corresponding bending-moment diagram are similar to those obtained by the model (see Fig. 6). See text
for further details.
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PA constriction increased RV diastolic pressure and,
as venous return to the LV was reduced, tended to
decrease LV diastolic pressure. The resultant changes
in TSP progressively flattened the septum and, in some
cases, produced a modest septal inversion in that it
became measurably concave with respect to the LV.
These rather subtle changes in the shape of the septum
should not be misconstrued when similar magnitudes
and temporal trends of TSP during PA constriction (see
Fig. 2) appear to have produced strikingly different
changes in curvature (1/R) (see Fig. 3).
For both the linear and nonlinear models, there was
a requirement that material stiffness be slightly
greater during PA constriction than for the corresponding control. It is unlikely that this difference in material stiffness resulted from changes in endosarcomeric
(e.g., titin) or extracellular matrix (e.g., collagen) proteins because the PA was constricted for ⬍30 s. It is
more likely that material stiffness increased owing to
an increase in coronary venous pressure. In isolated,
perfused dog hearts, Watanabe et al. (48) raised coronary venous pressure by increasing RA/RV pressure,
and showed LV end-diastolic pressure (LVEDP; an
indicator of LV stiffness) increased significantly at the
same LV volume. They also showed that coronary venous pressure was more effective in raising LVEDP
than coronary arterial pressure in the range tested.
They suggested that this apparent increase in stiffness
was related to an increase in intramyocardial blood
volume (26), as opposed to either increased pressure or
flow. Similarly, Resar et al. (34) have shown that transverse stiffness increased at least twofold as a result of
an increase in coronary arterial perfusion pressure
from 20 to 80 mmHg. Therefore, coronary venous congestion after PA constriction may explain the increase
in material stiffness predicted by our model.
The requirement for substantial end-bending moments during PA constriction (they were not required
during control) near end systole (Fig. 5) relative to the
rest of diastole is consistent with the septum being
much stiffer during this part of the cycle. Like the
pattern of time-varying elastance derived from a pressure-volume loop (42), G for both control and PA constriction decreased from end systole to a near-stable
value for the remainder of diastole (Fig. 4). Furthermore, the requirement for greater end-bending moments at the anterior insertion point (compared to the
posterior insertion point) reflects asymmetric flattening and inversion of the septum. Specifically, the curvature of the septum was greatest near the anterior
insertion point, and thus greater end-bending moments were required in order for conformity to be
obtained in this region of the septum. Whether this
asymmetry is due to anatomic differences in the two
insertion points or because of some artificially imposed
boundary (e.g., the dog was in the supine position
during echocardiographic recording) has yet to be determined.
The bending-moment diagrams shown in Fig. 6 illustrate that bending moment depends on the circumferential stress gradient, which develops across the struc-
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SEPTAL COMPRESSION DURING RV LOADING
able from that associated with myocardial ischemia.
This has usually been attributed to RV ischemia, despite the absence of coronary arterial stenoses. In addition, in the presence of left bundle branch block,
which also alters septal motion, thallium perfusion
studies are often “falsely positive,” suggesting relative
septal ischemia even though the coronary arteries are
shown to be normal (5). The demonstration of substantial circumferential compressive stresses (equal to
about ⫺20 mmHg) in the septum leads to a hypothesis
that must be tested rigorously: such stresses could
increase resistance to diastolic blood flow to a degree
sufficient to cause ischemia. Perfusion studies utilizing
either Doppler Flowire or contrast echocardiography
techniques will be required to test this hypothesis.
The desire to develop a model sufficiently complex to
detect structural differences in the septum between
control and PA constriction-loading conditions and yet
be practical resulted in the use of a 2-D FE model with
plane stress elements. Perhaps the most significant
limitation of such a model is that fibers, which render
the myocardium anisotropic, cannot be incorporated.
To attempt to account for the effect of myocardial
fibers, one could vary the stiffness in different directions across the mesh, i.e., incorporation of a parabolic
variation in circumferential elastic modulus (41). Varying the stiffness, however, would require more material parameters than for the isotropic neo-Hookean
hyperelasticity used here, making the problem more
difficult to solve numerically.
Another significant limitation pertains to the constitutive relation employed in our model. Many attempts
have been made to describe an appropriate constitutive
relation for the passive myocardium (21). As a first
approximation to the material properties of the septum, however, a neo-Hookean (i.e., linear stress-strain)
strain-energy function was used. Over the normal
working range of rat (38) and cat (9) trabeculae, SL
varies between 1.55 and 2.40 ␮m. Rodriguez et al. (35)
and Guccione et al. (19) have shown that SL varies
between 1.70 and 2.40 ␮m in the intact, normal canine
LV. Slack length of the sarcomere (SL0) has been
shown to be ⬃1.85 ␮m (44). Between SL of 1.60 ␮m
(15% negative strain) and 2.20 ␮m (20% positive
strain), the sarcomere force-length relationship is practically linear, i.e., the stiffness is nearly constant (38).
The stiffness increases steeply ⬍1.60 ␮m and ⬎2.20
␮m. Thus the material properties of the septum have
been approximated with the use of a neo-Hookean
strain-energy function over this range. Furthermore,
Rodriguez et al. (35) and Guccione et al. (19) showed
that SL ⫽ 2.20 ␮m occurred at an LVEDP of ⬃15
mmHg. Because LVEDP never exceeded 12 mmHg
under our control loading conditions, it is assumed that
SL did not exceed 2.20 ␮m. During PA constriction, the
maximum circumferential compressive strain achieved
at mid-diastasis was ⬎⫺15% (Fig. 7). The maximum
circumferential tensile strain achieved was ⬃10%.
We acknowledge Junyi Yi, Gerald Groves, and Cheryl Meek for
excellent technical assistance, and Linda Malley for echocardiographic recording.
This study was supported Medical Research Council of Canada
(Ottawa) Operating Grant MT-12886 (to J. V. Tyberg and N. G.
Shrive) and an accompanying Studentship Award (to G. S. Nelson).
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Study Limitations
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