Vol. 281, Issue 2, H506-H514, August 2001
Regional septal dysfunction in a
three-dimensional computational model of focal myofiber disarray
T. P.
Usyk,
J. H.
Omens, and
A. D.
McCulloch
Department of Bioengineering, Whitaker Institute for Biomedical
Engineering, University of California, La Jolla, California
92093-0412
 |
ABSTRACT |
MLC2v/ras transgenic mice
display a phenotype characteristic of hypertrophic cardiomyopathy, with
septal hypertrophy and focal myocyte disarray. Experimental
measurements of septal wall mechanics in ras transgenic mice
have previously shown that regions of myocyte disarray have reduced
principal systolic shortening, torsional systolic shear, and sarcomere
length. To investigate the mechanisms of this regional dysfunction, a
three-dimensional prolate spheroidal finite-element model was used to
simulate filling and ejection in the hypertrophied mouse left ventricle
with septal disarray. Focally disarrayed septal myocardium was modeled
by randomly distributed three-dimensional regions of altered material
properties based on measured statistical distributions of muscle fiber
angular dispersion. Material properties in disarrayed regions were
modeled by decreased systolic anisotropy derived from increased
fiber angle dispersion and decreased systolic tension development
associated with reduced sarcomere lengths. Compared with
measurements in ras transgenic mice, the model showed
similar heterogeneity of septal systolic strain with the largest
reductions in principal shortening and torsional shear in regions of
greatest disarray. Average systolic principal shortening on the right
ventricular septal surface of the model was 
0.114 for normal regions
and 0.065 for disarrayed regions; for torsional shear, these values were 0.047 and 0.019, respectively. These model results suggest that
regional dysfunction in ras transgenic mice may be explained in part by the observed structural defects, including myofiber dispersion and reduced sarcomere length, which contributed about equally to predicted dysfunction in the disarrayed myocardium.
myocardial mechanics; angular dispersion; sarcomere length; finite-element method; MCL2v/ras; familial hypertrophic
cardiomyopathy
| |
INTRODUCTION |
FOCAL MYOCYTE
DISARRAY is a common structural abnormality observed in many
cardiac diseases, including coronary heart disease, cor pulmonale, and
dilated cardiomyopathy (23a). It is particularly prevalent in patients
with hypertrophic cardiomyopathy (HCM), especially in the
interventricular septum, which commonly displays greater hypertrophy
than the free walls, with 25% or more of the wall disarrayed.
Because global ventricular performance can be apparently normal in HCM,
regional measures of myocardial function have been sought. Kramer et
al. (18) used magnetic resonance tissue tagging to measure
the distributions of systolic segment shortening in patients with HCM.
Compared with normal volunteers, circumferential and longitudinal
shortening strains were more heterogeneous in the patient group, with
the greatest depression in the septum. They concluded that this
heterogeneity of regional function might reflect the regional
distributions of myocardial disarray and fibrosis. In a similar study,
also using tagged magnetic resonance imaging, Dong et al.
(5) observed significantly decreased systolic wall
thickening and circumferential shortening strain in HCM patients compared with normal patients, with the greatest impairment of segment
function occurring in regions of the highest end-diastolic wall thickness.
While the presence of regional hypertrophy or disarray is associated
with altered distributions of segment shortening, establishing a direct
relationship between these structural abnormalities and altered
myocardial mechanics has necessitated measurements of regional strain
patterns in animal models. The MLV2v/ras transgenic mouse
was found to display focal myofiber disarray and hypertrophy, predominantly in the septal wall (11). Karlon et al.
(16) measured systolic septal wall strains and regional
myofiber angle distributions in hearts from these mice and compared
them with nontransgenic controls. Approximately 25% of the septal wall
had disarrayed myocytes, as characterized by a statistical dispersion of local fiber orientation about the mean >20°. Mean fiber angles were also different in regions of disarray. Systolic principal shortening and torsional shear strains on the right surface of the
septum were significantly lower in regions with underlying myocyte
disarray than the strains in nondisarrayed regions. Systolic and
diastolic sarcomere lengths were significantly shorter in disarrayed myocytes.
Although the study by Karlon and colleagues (16) showed
for the first time that reduced systolic strains do coincide with the
presence of focal alterations in myocardial tissue structure, it did
not establish whether the measured structural defects were themselves
directly responsible for the altered strain distributions. This does
seem likely. A change in the mean orientation of myofibers in areas of
disarray may alter local torsion because the transmural variation in
fiber orientation contributes to the development of twist during
systole. The increased angular deviation of disarrayed myofibers could
also reduce tissue anisotropy, which is necessary for the development
of systolic torsion (1). Lower sarcomere lengths suggest
that myocytes in areas of disarray may operate at a lower point on
their isometric length-tension curve, generating less systolic tension
and shortening. To test these hypotheses and determine the relative
importance of these potential mechanisms, we used three-dimensional
computational models to investigate septal wall mechanics in normal and
ras transgenic mouse left ventricles (LVs). Ventricular
geometry and fiber architecture, including the presence of randomly
distributed regions of disarray with shorter sarcomere lengths, were
modeled based on the experimental measurements reported previously
(13, 14).
Comparing model predictions with measured septal strain distributions
showed that the loss of regional shortening and shear in disarrayed
regions can be reasonably well explained by the observed focal defects
in myocyte orientation and sarcomere length, with each mechanism
contributing approximately equally to the observed loss of regional
segment function.
| |
METHODS |
Structural model.
Thick-walled ellipsoids were used to model the normal and hypertrophied
mouse LV. Wall thicknesses were matched to the septal thicknesses in
wild-type and ras transgenic mice as reported by Gottshall
et al. (6) using M-mode echocardiography: 0.7 ± 0.0 mm for wild-type mice and 0.9 ± 0.1 mm for ras
transgenic mice. The three-dimensional computational model with
disarray was represented in prolate spheroidal coordinates (
,
M and 
) using 1,024 trilinear finite elements and 1,360 nodes. Epicardial nodes at the base of the finite-element mesh were
fixed in the circumferential () and radial () directions to
simulate constraints imposed by the relatively stiff mitral valve
annulus (7); the longitudinal coordinate (M)
was fixed at all basal (M = 120°) and apical
(M = 5°) nodes. Transmural mean fiber orientations
were included in the model based on measurements made in control and
ras transgenic mouse hearts (16), using linear
interpolation of fiber angles between nodes. Sarcomere lengths in the
unloaded reference state were included based on previous measurements
(21).
Passive mechanical properties.
Stress and strain in the LV were modeled using the finite-element
method by passive filling to an end-diastolic pressure of 10 mmHg. An
exponential form of the strain energy function has been previously used
to model the resting myocardium (9, 23). The following
form of the strain energy function (W) assumes that the
material is hyperelastic, nearly incompressible, and orthotropic with
respect to fiber and laminar sheet axes (4), consisting of
the fiber axis Xf (along the myofibers), the
sheet axis Xs (perpendicular to
Xf and parallel to the sheet plane), and the sheet-normal axis Xn (perpendicular to the
laminar sheet plane)
|
(1)
|
where
|
(2)
|
where Eij are components of Green's
strain tensor E in an orthogonal coordinate system having
fiber, sheet, and sheet-normal (f, s, n) axes, respectively;
i and j = f, s, or n; and J is
the determinant of the stretch tensor U. The values of the material constants (Ccompr and
bij) have been previously estimated for
models of passive ventricular mechanics (23, 24) and
indicate that the normal passive myocardium is stiffer in the fiber
direction than in the transverse direction, consistent with
experimental measurements (19).
Active contraction model.
Systolic contraction was modeled by defining the Cauchy stress tensor
(T) referred to the local fiber coordinates as the sum of
the passive three-dimensional stress tensor
[T(p)] derived from the strain energy function
and an active stress tensor [T(a)]
|
(3)
|
The components
[T
] of the active
stress tensor were derived from the diagonal stress tensor (Tactive) referred to fiber coordinates using a
rotation matrix (q), which rotates a vector in the plane of
the wall through a (deformed) fiber angle about the radial axis
|
(4)
|
The components of the active tensor
Tactive were a function of peak intracellular
calcium concentration ([Ca2+]i) and sarcomere
length. The formulation of the active model previously developed
(8, 10, 12) was based on experimental measurements of
sarcomere length-tension relations in rat trabeculae. The transverse
components of the active stress tensor
[T
and
T
] were computed as a function
of the fiber component [T
] of
active stress as described by Usyk et al. (23) and Mazhari
at al (20) to be consistent with biaxial experimental
tests in barium-contracted rabbit myocardium (19).
Disarray was modeled in the septal area [circumferential coordinate:
(0°, 180°); longitudinal coordinate: M (30°, 110°)] by separate groups of finite elements. The pattern of
disarray was reconstructed from the regional distributions of disarray observed by Karlon et al. (16), who measured local fiber
orientation and angular deviation in 21 × 21 grids from each of
nine transmural tissue sections spaced equally through the septal wall
thickness. With the use of these original data, we calculated, for each
heart, the number of disarrayed three-dimensional regions and the
distribution of their volumes in seven mice. On the basis of these
statistics, we built a stochastic region-growing algorithm to generate
random three-dimensional regions of focal disarray in the model,
occupying ~25% of the finite elements (100 elements) in the septal
wall of the model (see Fig. 1 and
RESULTS). Each finite element in the septal wall of the
model was assumed to be either entirely normal or entirely disarrayed
(piecewise constant variation). The steps in generating a
region-growing algorithm are as follows: 1) Randomly select
the first element of region k by coordinates X
(1),
where i = 1 ... 3 and
X(j) = 1 ... Xi (max). 2)
If k > 1 and this element already exists or is
connected with an existing region, then repeat step 1. 3) Randomly select the direction of growth and, using this
direction, define the coordinates of a new element
(j) of region k. 4) If
X(j) > Xi (max), or if
X < 1, or if
this element already exists or is connected with an existing region,
then repeat step 3. 5) Randomly select an element
in the current region (k) for the next step. 6)
j = j + 1. 7) Repeat steps 3-6 while j 
j
, where j is the maximal number of
elements in region k. 8) k = k + 1. 9) Repeat steps 1-8
while k kmax, where
kmax is the maximal number of regions.
View larger version (41K):
[in this window]
[in a new window]
|
Fig. 1.
Fiber orientation and disarray location for the left
ventricular (LV) endocardial surface (A) and right side of
the septum (B).
|
|
Local mean myofiber orientations in histological tissue sections from
ras transgenic mouse hearts were measured previously using
automated methods (14). Mean fiber orientations for normal mice (without disarray) and ras transgenic mice were
calculated. On the right ventricular (RV) side of the septum, mean
fiber angles were approximately 75° in control mice and 60° for
ras transgenic mice. On the LV side, these angles were
+75° and +45°, respectively. There were no significant differences
between mean fiber angles in normal and disarrayed tissues of
ras transgenic mice (14).
The influence of two different mechanisms associated with
microstructural alterations of disarrayed tissue were examined with the
model: 1) local increase in myofiber angular dispersion (AD) and 2) local decrease in sarcomere length. Each of these
structural alterations were modeled individually by altering material
properties in the finite-element model.
Mechanism 1: increase in myofiber AD.
Dispersion of fiber orientation was measured by Karlon et al.
(16) as the local AD (angular equivalent of standard
deviation) of myofiber orientation. To determine the net contribution
of a family of myofibers with a known distribution of orientations, a
superposition of local stress contributions was performed. Fiber, sheet, and sheet-normal stresses acting along the mean myofiber, sheet,
and sheet-normal axes in an ensemble of cells is obtained by
integrating myofibril tractions over the distribution of myofiber angles f(
)
|
(5)
|
where T
is a function
of peak [Ca2+]i and sarcomere length;
T
and
T
are constant fractions
(k) of the active fiber stress
T (23); and µ are angles that describe the relationship between the local
myofiber axis and the mean fiber axis; f() is the fiber
orientation probability density distribution, which can be described
using a von Mises distribution; 
(µ) is also density distribution,
which can be described as (µ) = 1/(2
) [µ (1;2)]; and the rotation matrix Q defines the relation between the mean fiber-sheet coordinate system
(Xfiber-sheet) and local cell coordinate system
(Xlocal)
|
(6)
|
|
(7)
|
where fo() is the fiber
orientation probability density in the undeformed reference state and
i are fiber, sheet, and normal-sheet
extensions. The function
fo() may also depend on the angle
µ.
We can rewrite the equation for Tactive as
follows
|
(8)
|
After integration, we have
|
(9)
|
With the use of these equations, with a measured average AD of
12° for normal myocardium and 25° for disarrayed tissue, we obtained a ratio of systolic cross-fiber to fiber stress of 0.452 in
normal muscle versus 0.697 in disarrayed elements.
Mechanism 2: reduction in sarcomere length.
Development of systolic tension in cardiac myocytes is dependent on
activator calcium concentration and sarcomere length (22a). The
isometric length-tension curve describes the tension development at
different sarcomere lengths for given concentrations of calcium. This
suggests that these myocytes operate at a lower point of the sarcomere
length-tension curve, thus generating less tension and shortening.
Systolic isometric fiber stress development in the model was computed
as a function of activator calcium concentration and sarcomere length
according to the model of length-dependent activation by Guccione et
al. (9).
Areas of disarray were modeled with sarcomere lengths measured in the
reference state (13). Briefly, groups of ras
transgenic (n = 6) and control hearts
(n = 7) were fixed at zero pressure and under barium
contracture as a model of end systole. The tissue was embedded in
plastic, sectioned at a thickness of 5 µm, and stained with toluidine
blue. Sarcomere lengths were found at ×1,000 magnification by
measuring 10 adjacent sarcomere lengths in at least 10 random locations
in any given section. Measurements were made at subepicardial
(~15-25% depth), midwall (45-55% depth), and
subendocardial (75-85% depth) regions. Measurements were made in
five sections from each depth each with a different knife angle. The
smallest average sarcomere length was used from each depth because
off-angle sectioning resulted in overestimation of the measurements.
Sarcomere lengths were significantly smaller in areas of myofiber
disarray compared with areas of normally arrayed tissue in ras
transgenic mouse hearts and compared with control hearts in both
the zero-pressure and barium contracture states. There was no
significant difference between the control group of hearts and areas of
normally arrayed tissue in ras transgenic mouse hearts at
either loading state. No significant variation with wall depth was
found in sarcomere length at either zero-pressure or under barium
contracture in either group of animals. Thus we used the measured
midseptal values for average resting sarcomere length in the model
(2.00 µm for control and 1.65 µm for disarray).
| |
RESULTS |
Biaxial model of disarrayed tissue.
The biaxial contraction of orthotropic cubes with material properties
of normal cardiac muscle or disarrayed myocardium was simulated. Figure
2 shows stress-strain curves in fiber and
cross-fiber directions for normal tissue and disarrayed tissue. The
influence of reduced sarcomere length and myofiber AD are also shown
separately. These results illustrate the significant effects of
disarray on myocardial systolic mechanical properties.
View larger version (19K):
[in this window]
[in a new window]
|
Fig. 2.
Systolic stress-strain curves for normal and disarrayed
tissue in biaxial deformation of an orthotropic cube. The influence of
sarcomere length (SL), angular dispersion (AD), and both of these
factors together (SL + AD) are shown. A: fiber
stress-strain relations. B: cross-fiber stress-strain
relations.
|
|
LV model of a wild-type mouse heart.
Figure 3 compares model results with
experimentally measured data (14) for the principal
strains E1 and E2 and the
torsional shear E12 on the RV surface of the
septum in the wild-type mouse heart. Computed strain components for the
LV surface of the septum are also shown. On the RV surface of the
septum, the model showed good agreement with measurements of
E1 and E12 but poorer
agreement for E2.
View larger version (18K):
[in this window]
[in a new window]
|
Fig. 3.
Average first principal strain E1,
second principal strain E2, and torsional shear
E12 found on the right ventricular (RV) surface
of the septum and on the LV surface of the septum [model and
experimental data (14)].
|
|
Effect of septal wall hypertrophy.
Figure 4 shows strain components computed
with the hypertrophied but not disarrayed LV model normalized to the
corresponding strains found in model of the wild-type mouse heart.
There was an increase in the septal in-plane shear
E12, especially on the RV surface, and a
decrease in the second principal strain E2. The
major principal strain E1 did not change
significantly. These effects of hypertrophy are consistent with
calculated results for a two-dimensional model (14) and
previous experimental investigations (11).
View larger version (17K):
[in this window]
[in a new window]
|
Fig. 4.
Average strain for model of hypertrophied mouse LV
normalized to strains found in the wild-type mouse model. First
principal strain E1, second principal strain
E2, and torsional shear
E12 found on the RV surface and the LV surface
of the septum are shown.
|
|
Spatial distribution of disarray.
Table 1 shows average values for
the number and fractional volume of three-dimensional disarrayed
regions reconstructed from the original histological measurements
(16). The locations of septal disarray areas showed no
consistent pattern between individuals. It can be seen from this
statistical analysis that the disarrayed myocardium typically consisted
of one large region with a complicated geometry and several smaller
regions. With the use of these statistics on the number and size of
three-dimensional disarrayed regions, a random region-growing algorithm
was used to generate the pattern of disarray used in the model. Table 1
compares the summary statistics of the modeled distribution of disarray
with those computed from the original histological measurements.
View this table:
[in this window]
[in a new window]
|
Table 1.
Summary statistics of the modeled distribution of disarray with those
computed from original histological measurements
|
|
Wall thickening.
Wall thickening at end systole referenced to end diastole was
calculated for the normal, hypertrophic, and disarrayed hypertrophic walls. The hypertrophied wall was thicker at end diastole, but the
percent wall thickening was reduced from 41.0% to 29.6%. The change
in fiber orientation produced a reduction in wall thickening, whereas
addition of cross-fiber tension produced an increase in wall
thickening. Change in sarcomere length or a combination of all
mechanisms produced only a small reduction in wall thickening.
Strain and stress components.
Maps of the three-dimensional principal strain components
E1 and E2 and shear
strain E12 for a realistic distribution of
disarray are shown for the right side of the septum and the
subendocardial surface in Fig. 5. The
area shown is similar to the area where nonhomogeneous strains were
measured in isolated heart experiments (14). The maximum
principal strain E1 was most sensitive to the
presence of disarray, similar to a previous experimental finding (15). There was a local increase in the magnitude of
E1 associated with the increase in AD. The
influence of disarray on the strain components
E1 and E12 was more
significant at the right side of the septum than the LV subendocardium
(left side). Conversely, for the minimal principal strain component
E2, the influence of disarray was greater on the
left and less on the right side. A map of principal stress components
T1 and T2 for the right
and left side of the septum (Fig. 6)
shows that the effect of disarray was greater on
T1 than on T2.
View larger version (80K):
[in this window]
[in a new window]
|
Fig. 5.
Maps of principal strains E1 and
E2 and shear strain E12
for the LV endocardial surface (A) and right surface of the
septum (B).
|
|
View larger version (101K):
[in this window]
[in a new window]
|
Fig. 6.
Maps of principal stresses T1 and
T2 for the LV endocardial surface (A)
and right surface of the septum (B).
|
|
Average correlation coefficients were calculated to compare the local
RV septal surface strain with local AD values from four tissue sections
(in the middle of each finite element). The positive correlation
coefficient of E1 indicates a relationship
between increased AD and more positive E1
(reduced shortening). The correlation coefficient was significantly
greater for the section closest to the RV septal surface, indicating a
closer spatial relationship between disarray and septal surface
dysfunction. There was a trend toward more negative correlation near
the RV septal surface and positive correlation at the LV septal
surface. Because E2 was generally negative in
sign, with positive correlation near the LV septal surface, this
indicates that greater AD (increased disarray) is more likely to be
associated with more positive E2. Both these results indicate that the surface strain is reduced in areas of disarray, and both agree qualitatively with our earlier experimental investigations (14).
Average principal strains on the RV and LV septal surfaces
corresponding to areas of disarray are shown in Fig.
7. There was a significantly reduced
average systolic strain associated with disarrayed tissue found near
the RV septal surface. The subendocardial systolic strains associated
with disarrayed tissue found near the LV side of the septal wall were
less reduced. The average surface strains E2 and
E12 for areas of disarray are also shown in Fig.
7. There were significantly smaller average RV surface shear strains
E12 and principal strains
E2, associated with areas of disarrayed tissue
near the RV septal surface. Near the LV septal surface, torsional shear
was increased compared with the model of hypertrophy. The effects of
reduced sarcomere length and reduced anisotropy are separately shown in
Fig. 7 for principal strains E1 and
E2 and shear strain E12.
The contribution of reduced sarcomere lengths was similar to that of
decreased anisotropy for strains E1 and
E12 but different for E2
(see Fig. 7).
View larger version (19K):
[in this window]
[in a new window]
|
Fig. 7.
Average strains on the RV surface and LV surface of the
septum normalized to strains for the hypertrophied mouse model. First
principal strain E1, second principal strain
E2, and torsional shear
E12 in areas with disarray found at the
subepicardium (RV) or subendocardium (LV) are shown. The influence of
SL, AD, and both of these factors together (SL + AD) in the model
are shown.
|
|
Figure 8 shows average values of the RV
septal surface for E1 and
E12, associated with areas of the wall that are
affected by various amounts of disarray normalized to corresponding
strain solutions from the model of the wild-type mouse heart. Areas
with more dense disarray were associated with smaller RV septal surface shear strain E12, similar to the experimental
results (16), but the observed differences were greater
(see Fig. 8). Maximum and minimum principal systolic shortening
(E1 and E2) in more densely disarrayed areas were smaller than those in less dense areas,
but these variations were less significant than for the torsional
systolic shear strain E12.
View larger version (20K):
[in this window]
[in a new window]
|
Fig. 8.
Average strains on the RV surface of the septum
normalized to strains found in the wild-type mouse model. First
principal strain E1 and torsional shear
E12 associated with varying amounts of septal
wall disarray for the model and experimental observations
(14) are shown.
|
|
| |
DISCUSSION |
We used a numerical model to test the hypothesis that focal
changes in the microstructural properties of the disarrayed myocardium are directly responsible for the patterns of regional dysfunction that
were recently measured on the septal surface of the
MLC2v/ras transgenic mouse heart (16). We
investigated the influence of three different structural properties of
disarrayed myocardium using a three-dimensional finite-element model of
systolic contraction. Models were created of the normal and
hypertrophied ventricle based on previously published measurements made
on the transgenic mouse with ventricular expression of ras,
which displays morphological characteristics of HCM (11).
A realistic three-dimensional distribution of disarray affecting
~25% of the wall was used to match the experimental measures.
The model of the wild-type mouse heart did not agree especially well
with experimental measurements, especially for the second principal
strain (see Fig. 3). This may be because, for the present study, a
simple ventricular geometry was used and material properties were
assumed to be transversely isotropic, neglecting the possible effects
of myocardial laminar sheet structure. The coefficients of the
constitutive law were chosen from previous studies (23, 24) in other species, which may not be representative of the mouse heart, especially under conditions of hypertrophy and disarray. Indeed, focal fibrosis is commonly observed to accompany myocyte disarray in animals and humans and was observed in the ras
transgenic mouse. This may increase regional diastolic muscle stiffness
and possibly effect systolic mechanics as well. We examined the
sensitivity of the model results to passive material parameters by
repeating the analysis with a different set of constitutive parameters
derived from recent experiments in mice. The following material
parameters of a transversely isotropic incompressible constitutive
equation were estimated from uniaxial tests in passive murine papillary muscles and isolated ventricles: C1 = 0.126, b1 = 47.94, b2 = 9.59, and
b3 = 47.94 (S. M. Weis, J. H. Omens, and A. D. McCulloch, unpublished observations).
These coefficients are significantly different than those used in
current study (C was 86% lower and bii were 160-2,200% greater). To determine
how these differences in passive constitutive parameters may influence
systolic strain results in the model, additional numerical analyses
were obtained using these parameters. We found that these large
differences in passive parameters nonetheless had only a small effect
on computed systolic strains. Average E1
decreased 4%, E2 decreased 7%, and E12 increased 8% of their corresponding values;
these are very small differences in terms of absolute strain. Thus,
with the current systolic model, changes in the passive material
properties do not have a large influence on systolic strains. Although
changes in tissue ultrastructure are known to occur in this mouse
model, for example, increased collagen content, we did not take these into account, but we conclude that passive tissue alterations will not
substantially affect these systolic results. In the present numerical
study of septal disarray, sarcomere disarray within myocytes, which has
been observed pathologically, was also neglected. This assumption could
also influence sarcomere function independent of calcium level or
sarcomere length, for example, it could lead to a reduction in the
efficiency of the contractile function. Reduced sarcomere reference
length may not be the only mechanism responsible for altered systolic
performance, because studies have shown altered calcium levels as well
as altered calcium sensitivity of sarcomere proteins in animal models
of HCM (15). These changes were not taken into account
with the present modeling approach.
The effect of hypertrophy of the wall was examined by increasing wall
thickness from 0.7 to 0.9 mm based on published septal wall
measurements made using M-mode echocardiography (11).
There was an increase in all systolic epicardial torsional strain and a
decrease in the second principal strain component. Because there is an
increase in ventricular mass relative to the inner diameter, it was
expected that a greater number of fibers would generate a larger amount
of tension, thus generating greater torsion at the RV septal surface.
Also, the increase in size was made by extending the outer wall but
keeping the cavity the same size as the normal model. This had the
additional effect of increasing the radius of the epicardial fibers,
offering a mechanical advantage. Taber et al. (22)
suggested that the greater mechanical advantage of epicardial fibers is
one possible mechanism for the positive epicardial shear strain and
twist found in systole consistent with a dominance of epicardial fibers
over endocardial fibers (22). The different behavior of
torsional shear on the RV and LV surfaces of the septum (see Fig. 7)
can probably be explained by the different fiber orientations on either
side of the septum. This difference could also be due, in part, to
changes in the activation sequence from the LV endocardium to the RV
septal surface or to the impact of myofiber disarray on conduction velocity.
The influence of decreased anisotropy and reduced sarcomere length were
almost the same on the maximal principal shortening and torsional
strain but different for the minimum principal shortening. Because
the experimental studies showed no significant changes in
E2, we focused our comparisons primarily on the
other strain components. A combination of all characteristics of the
disarrayed myocardium resulted in the greatest effect on strain and the
best agreement between model and experimental observations.
The greatest effect of myofiber disarray on experimental epicardial
function was found for the maximum principal strain
E1 and torsional shear (15). This
finding was also duplicated in the present finite-element modeling
study. It has been suggested by Lin and Yin (19) that the
dispersion of fibers in the normal myocardium may be at least partly
responsible for the development of substantial cross-fiber systolic
stress. Bonvendeerd et al. (3) showed that myofiber
imbrication angle causes local variation in strain magnitudes in a
model of the LV. These investigations highlight the importance of
myofiber dispersion in regional mechanics. Imbrication angle was not
included in our analysis but is typically <5°.
More extensive disarray had a greater influence on epicardial strain
than less extensive disarray. Additionally, this result indicates that
disarray that affects a greater percentage of the wall thickness has
the greatest effect on strain. This finding is also consistent with
experimental results (14) showing that when a greater
percentage of the wall thickness was affected by disarray, there was a
greater reduction in RV surface strain components. The differences in
strain components between the model and experimental observations for
the same amounts of disarray were most significant for the shear strain
(see Fig. 8). The change in material properties between disarrayed and
nondisarrayed tissue was modeled sharply from element to element, which
is unlikely to approximate the transition in real tissue and might be
one explanation for the differences between numerical and experimental
results. This may also explain why the torsional shears for the
nondisarrayed tissue were greater than those in the model of the
wild-type mouse heart, which also disagrees with the experimental findings.
Experimental evidence from magnetic resonance imaging studies on human
HCM is inconclusive, with some investigators finding a decrease in wall
thickening associated with the hypertrophic septum (5) but
others finding no change (26). Echocardiographic studies
on the ras transgenic mouse model did not show evidence of
alteration in wall thickening (11).
This finite-element study indicates that multiple mechanisms may be
responsible for the dysfunction associated with myofiber disarray and
may be associated with diseases such as familial HCM. The findings from
the finite-element model are qualitatively similar to those found in
experimental studies. We conclude that decreased fiber tension
associated with increased AD and reduced sarcomere length contribute
approximately equally to the dysfunction in disarrayed regions.
| |
ACKNOWLEDGEMENTS |
We thank Dr. W. Karlon for valuable advice and contributions to
this work.
| |
FOOTNOTES |
This research was supported by National Science Foundation Grant
BES-9634974 and the National Biomedical Computation Resource (National
Institutes of Health Grant RR-08605).
Address for reprint requests and other correspondence: A. D. McCulloch, UCSD, Dept. of Bioengineering, 9500 Gilman Dr., La Jolla,
CA 92093-0412 (E-mail: amcculloch{at}ucsd.edu).
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.
Received 21 December 2000; accepted in final form 14 March 2001.
| |
REFERENCES |
1.
Arts, T,
Hunter WC,
Douglas AS,
Muijtjens AM,
Corsel JW,
and
Reneman RS.
Macroscopic three-dimensional motion patterns of the left ventricle.
Adv Exp Med Biol
346:
383-392,
1993[Medline].
3.
Bovendeerd, PH,
Huyghe JM,
Arts T,
van Campen DH,
and
Reneman RS.
Influence of endocardial-epicardial crossover of muscle fibers on left ventricular wall mechanics.
J Biomech
27:
941-951,
1994[Web of Science][Medline].
4.
Costa, KD,
Takayama Y,
McCulloch AD,
and
Covell JW.
Laminar fiber architecture and three-dimensional systolic mechanics in canine ventricular myocardium.
Am J Physiol Heart Circ Physiol
276:
H595-H607,
1999[Abstract/Free Full Text].
5.
Dong, SJ,
MacGregor JH,
Crawley AP,
McVeigh E,
Belenkie I,
Smith ER,
Tyberg JV,
and
Beyar R.
Left ventricular wall thickness and regional systolic function in patients with hypertrophic cardiomyopathy. A three-dimensional tagged magnetic resonance imaging study.
Circulation
90:
1200-1209,
1994[Abstract/Free Full Text].
6.
Gottshall, KR,
Hunter JJ,
Tanaka N,
Dalton N,
Becker KD,
Ross J,
and
Chien KR.
Ras-dependent pathways induce obstructive hypertrophy in echo-selected transgenic mice.
Proc Natl Acad Sci USA
94:
4710-4715,
1997[Abstract/Free Full Text].
7.
Guccione, JM,
Costa KD,
and
McCulloch AD.
Finite element stress analysis of left ventricular mechanics in the beating dog heart.
J Biomech
28:
1167-1177,
1995[Web of Science][Medline].
8.
Guccione, JM,
and
McCulloch AD.
Mechanics of active contraction in cardiac muscle. I. Constitutive relations for fiber stress that describe deactivation.
J Biomech Eng
115:
72-81,
1993[Web of Science][Medline].
9.
Guccione, JM,
McCulloch AD,
and
Waldman LK.
Passive material properties of intact ventricular myocardium determined from a cylindrical model.
J Biomech Eng
113:
42-55,
1991[Web of Science][Medline].
10.
Guccione, JM,
Waldman LK,
and
McCulloch AD.
Mechanics of active contraction in cardiac muscle: part II-cylindrical models of the systolic left ventricle.
J Biomech Eng
115:
82-90,
1993[Web of Science][Medline].
11.
Hunter, JJ,
Tanaka N,
Rockman HA,
Ross J,
and
Chien KR.
Ventricular expression of a MLC-2v-ras fusion gene induces cardiac hypertrophy and selective diastolic dysfunction in transgenic mice.
J Biol Chem
270:
23173-23178,
1995[Abstract/Free Full Text].
12.
Hunter, PJ,
McCulloch AD,
and
ter Keurs HE.
Modelling the mechanical properties of cardiac muscle.
Prog Biophys Mol Biol
69:
289-331,
1998[Web of Science][Medline].
13.
Karlon, WJ.
Influence of Myocardial Fiber Organization on Ventricular Function. San Diego, CA: University of California San Diego Press, 1998.
14.
Karlon, WJ,
Covell JW,
McCulloch AD,
Hunter JJ,
and
Omens JH.
Automated measurement of myofiber disarray in transgenic mice with ventricular expression of ras.
Anat Rec
252:
612-625,
1998[Medline].
15.
Karlon, WJ,
Hsu PP,
Li S,
Chien S,
McCulloch AD,
and
Omens JH.
Measurement of orientation and distribution of cellular alignment and cytoskeletal organization.
Ann Biomed Eng
27:
712-720,
1999[Web of Science][Medline].
16.
Karlon, WJ,
McCulloch AD,
Covell JW,
Hunter JJ,
and
Omens JH.
Regional dysfunction correlates with myofiber disarray in transgenic mice with ventricular expression of ras.
Am J Physiol Heart Circ Physiol
278:
H898-H906,
2000[Abstract/Free Full Text].
18.
Kramer, CM,
Reichek N,
Ferrari VA,
Theobald T,
Dawson J,
and
Axel L.
Regional heterogeneity of function in hypertrophic cardiomyopathy.
Circulation
90:
186-194,
1994[Abstract/Free Full Text].
19.
Lin, DH,
and
Yin FC.
A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture or tetanus.
J Biomech Eng
120:
504-517,
1998[Web of Science][Medline].
20.
Mazhari, R,
and
MAD
Three-dimensional mechanics of myocardial contraction: mechanisms of transverse systolic stress. Proceeding of ASME 4th Summer Bioengineering Conference, 1999, p. 43-44.
21.
Rodriguez, EK,
Omens JH,
Waldman LK,
and
McCulloch AD.
Effect of residual stress on transmural sarcomere length distributions in rat left ventricle.
Am J Physiol Heart Circ Physiol
264:
H1048-H1056,
1993[Abstract/Free Full Text].
22.
Taber, LA,
Yang M,
and
Podszus WW.
Mechanics of ventricular torsion.
J Biomech
29:
745-752,
1996[Web of Science][Medline].
22a.
Ter Keurs, HE,
Rijnsburger WH,
van Heuningen R,
and
Nagelsmit MJ.
Tension development and sarcomere length in rat cardiac trabeculae. Evidence of length-dependent activation.
Circ Res
46:
703-714,
1980[Free Full Text].
23.
Usyk TP, Mazhari R, and McCulloch AD. Effect of laminar
orthotropic myofiber architecture on regional stress and strain in the
canine left ventricle. J Elasticity. In press.
23a.
Van der Bel-Kahn, J.
Muscle fiber disarray in common heart disease.
Am J Cardiol
40:
355-364,
1977[Web of Science][Medline].
24.
Vetter, FJ,
and
McCulloch AD.
Three-dimensional stress and strain in passive rabbit left ventricle: a model study.
Ann Biomed Eng
28:
781-792,
2000[Web of Science][Medline].
26.
Young, AA,
Kramer CM,
Ferrari VA,
Axel L,
and
Reichek N.
Three-dimensional left ventricular deformation in hypertrophic cardiomyopathy.
Circulation
90:
854-867,
1994[Abstract/Free Full Text].
Am J Physiol Heart Circ Physiol 281(2):H506-H514
0363-6135/01 $5.00
Copyright © 2001 the American Physiological Society
TOP
ABSTRACT
INTRODUCTION
