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Ultrasonic determination of the anisotropy of Young’s modulus of fixed tendon and
fixed myocardium
Brent K. Hoffmeister, Scott M. Handley, Samuel A. Wickline, et al.
Citation: The Journal of the Acoustical Society of America 100, 3933 (1996); doi: 10.1121/1.417246
View online: https://doi.org/10.1121/1.417246
View Table of Contents: https://asa.scitation.org/toc/jas/100/6
Published by the Acoustical Society of America
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Ultrasonic determination of the anisotropy of Young’s modulus
of fixed tendon and fixed myocardium
Brent K. Hoffmeister, Scott M. Handley, Samuel A. Wickline, and James G. Miller
Washington University, Department of Physics and School of Medicine, St. Louis, Missouri 63130
~Received 3 August 1995; accepted for publication 19 June 1996!
The linear elastic properties of a soft tissue exhibiting a unidirectional arrangement of reinforcing
fibers may be described in terms of the five independent elastic stiffness coefficients c11,c13,c33,
c44, and c66. In previous studies, ultrasonic measurements of these coefficients for formalin fixed
specimens of bovine Achilles tendon and normal human myocardium were reported. In the present
study these results are used to analyze the anisotropy of Young’s modulus of these tissues. For
formalin fixed tendon a value of 1.37 GPa is obtained for Young’s modulus along the fiber axis of
the tissue, and a value of 0.0706 GPa is obtained perpendicular to the fibers. For formalin fixed
myocardium, values of 0.101 and 0.0311 GPa parallel and perpendicular to the fibers, respectively,
are obtained. Based on the results for the angular dependence of Young’s modulus from
unidirectional specimens of myocardium, a model is introduced to estimate these features for the
more complicated fiber architecture of the left ventricular wall. © 1996 Acoustical Society of
America.
PACS numbers: 43.80.Cs, 43.80.Ev @FD#
INTRODUCTION
The linear elastic behavior of a soft tissue exhibiting a
unidirectional arrangement of reinforcing fibers with a ran-
dom transverse distribution may be described in terms of the
five independent elastic stiffness coefficients c11,c13,c33,
c44, and c66. In previous studies, we reported ultrasonic mea-
surements of these coefficients for formalin fixed specimens
of bovine Achilles tendon and normal human myocardium.
Three of these coefficients ~c11,c33, and c13!were deter-
mined by measuring the velocity of quasilongitudinal mode
ultrasonic pulses as a function of angle of propagation rela-
tive to the fiber axis of these tissues.1–3 The remaining two
coefficients were determined by measuring the velocity and
attenuation of transverse mode ultrasonic waves propagated
perpendicular to the fiber axis with the polarization oriented
either parallel ~c44!or perpendicular ~c66!to the fibers.4
In general, the fiber architecture of normal human myo-
cardium is more complicated than a simple unidirectional
arrangement of myofibers. For example, histologic studies of
the left ventricular wall by Streeter et al. have established
that the myofibers exhibit a progressive transmural shift in
orientation from approximately 160° at the endocardial sur-
face to 260° at the epicardial surface.5To facilitate a de-
scription of the elastic properties of myocardium in terms of
five independent elastic stiffness coefficients, the specimens
of myocardium considered in this study were prepared from
the left ventricular wall in a fashion that permitted the ultra-
sonic interrogation of thin regions of tissue located at spe-
cific transmural sites.1,4 The predominant fiber orientation of
these regions was unidirectional.
In contrast to myocardium, the Achilles tendon exhibits
a unidirectional arrangement of reinforcing fibers throughout
its entire thickness. The Achilles tendon may be qualitatively
described as a unidirectional arrangement of long stiff fibers
of collagen embedded in an amorphous gel-like matrix. The
fibers themselves exhibit a fairly well defined hierarchical
organization in which microfibrils of collagen are arranged
to form subfibrils which in turn are assembled to form col-
lagen fibrils. In tendon, the fibrils are organized into a final
structural unit called the fascicle which has a triangular cross
section approximately 150 to 300
m
m in diameter.6
The concentration of interstitial collagen may represent
an important difference between tendon and myocardium.
Collagen is a structural protein that generally occurs in high
concentrations in certain connective tissues such as tendons.
Collagen constitutes approximately 85% of the mass of most
tendons as measured by dry weight ~30% by wet weight!.7–9
Myocardium also contains collagen, but to a lesser degree.
The approximate total content of collagen in normal human
myocardium is 4% by dry weight ~1% by wet weight!.7,8,10
Despite its modest concentrations in the cardiac interstitium,
collagen has been identified as a potentially significant deter-
minant of the structural properties of myocardium.11
Alterations in the concentration and organization of
myocardial collagen often correlate with cardiac
dysfunction.12 For example, left ventricular pressure over-
load hypertrophy may produce an increase in collagen con-
centration and a structural remodeling of the connective tis-
sue matrix of the left ventricle. This accumulation of
collagen ~fibrosis!may also result from systemic hyperten-
sion. Local increases in collagen concentration often accom-
pany myocardial infarction. Because myocytes cannot regen-
erate, collagen replaces the necrotic tissue with a stiff patch
of scar tissue to maintain the mechanical integrity of the
ventricle. After myocardial infarction, the injured region may
accumulate collagen up to five times that of noninfarcted
regions.11
In a recent study using ultrasonic backscatter measure-
ments to investigate the fiber architecture of regions of ma-
ture infarction in explanted human hearts, our laboratory re-
ported that the scar tissue manifested a progressive
transmural shift in fiber orientation similar to that of normal
3933 3933J. Acoust. Soc. Am. 100 (6), December 1996 0001-4966/96/100(6)/3933/8/$10.00 © 1996 Acoustical Society of America
myocardium.13 This preservation of the fiber architecture of
the ventricular wall by the scar tissue may be a response to
uniaxial stresses imposed at each transmural level by the
contraction of surrounding normal myocardium. At a given
transmural level the scar tissue fibers remain unidirectionally
aligned, and serve to mechanically couple the contraction of
the surrounding normal myocytes. In this sense, the scar tis-
sue functions in an analogous fashion to tendons that couple
the uniaxial tensile forces generated by the contraction of
skeletal muscles to bone. Consequently, tendon may offer a
useful biological model for estimating some of the mechani-
cal characteristics of myocardial fibrosis.
In the present study, we examine some of the anisotropic
mechanical features of formalin fixed specimens of tendon
and myocardium using our reported measured values for the
elastic stiffness coefficients of these tissues to determine
Young’s modulus in all spatial directions. Our approach is
based on the linear theory of elasticity, which assumes that
the strain induced in the tissue varies linearly with the ap-
plied stress. This is generally accepted as a reasonable ap-
proach considering the small range of strains induced by ul-
trasonic waves. However, this assumption of linearity is not
necessarily valid over the much larger range of strains that
may be induced by normal physiologic stresses in vivo.
I. ANISOTROPY OF YOUNG’S MODULUS
To extract values of Young’s modulus from the elastic
stiffness coefficients, we analyzed the following elastic stiff-
ness matrix appropriate for a material exhibiting a unidirec-
tional arrangement of reinforcing fibers with a random trans-
verse distribution
@cIJ#5
3
c11 c1122c66 c13 000
c
1122c66 c11 c13 000
c
13 c13 c33 000
000c
44 00
0000c
44 0
0 0 000c
66
4
,
~1!
where the 3 axis is along the fiber axis of the material. For
the present study we computed Young’s modulus as a func-
tion of angle ~of the stress axis!in 1° increments relative to
the fiber axis of the tissue. This was accomplished by per-
forming a Bond rotation14 to position the 3 axis along each
direction of interest. The elastic compliance matrix, [sIJ],
was determined by numerically inverting the rotated elastic
stiffness matrix, and the value for Young’s modulus was
obtained from the reciprocal of the compliance coefficient
s33.All values reported in this study for Young’s modulus
were obtained by the numerical means just described. How-
ever, it is also possible to obtain a general expression for
Young’s modulus as a function of the elastic stiffness coef-
ficients and angle of the stress axis relative to the fiber axis
of the tissue. Along certain directions of high symmetry this
expression takes on a relatively compact form. For a material
whose linear elastic properties are described by the form of
the stiffness matrix in Eq. ~1!, Young’s modulus parallel and
perpendicular to the fiber axis ~the 3 axis!may be expressed
as
Eparallel5c332c13
2
c112c66 ~2!
and
Eperpendicular52c66
F
11c33~c1122c66!2c13
2
c11c332c13
2
G
.~3!
Values for the elastic stiffness coefficients of formalin
fixed specimens of bovine Achilles tendon and normal hu-
man myocardium were obtained from previously reported
ultrasonic studies of these tissues.1–4 These values are repro-
duced in Table I. For the present study these values were
used to compute the angular dependence of Young’s modu-
lus for each tissue. Our results are illustrated in Fig. 1 for the
meridian plane. The meridian plane is defined as any plane
containing the fiber axis of the material. The upper panel of
this figure shows Young’s modulus plotted as a function of
angle relative to the fiber axis of formalin fixed bovine
Achilles tendon. Fixed tendon demonstrates a substantial an-
isotropy in Young’s modulus with two peaks occurring 180°
apart. These maxima occur for angles parallel to the fiber
axis of the tissue. Minima occur for angles perpendicular to
the fiber axis. An interesting feature of this plot is the rapid
variation in Young’s modulus for angles near 0° and 180°.
This feature is characteristic of many of the unidirectional
graphite fiber reinforced composites our laboratory has in-
vestigated in other studies.15,16
The lower panel of Fig. 1 illustrates our results for for-
malin fixed normal human myocardium. Similar to fixed bo-
vine tendon, fixed human myocardium exhibits a rapid varia-
tion in Young’s modulus near 0° and 180° with the maxima
occurring parallel to the fiber axis of the tissue. However, it
TABLE I. Summary of measured results for the elastic stiffness coefficients of formalin fixed bovine Achilles
tendon and formalin fixed normal human myocardium. The five independent coefficients presented in this table
characterize the linear elastic properties of specimens of tissue exhibiting a unidirectional arrangement of
reinforcing fibers aligned along the 3 axis.
Tissue c11
~GPa!
c33
~GPa!
c44
~GPa!
c66
~GPa!
c13
~GPa!
Formalin fixed bovine
Achilles tendon 3.08 4.51 3.72310221.80310223.10
Formalin fixed normal
human myocardium 2.46 2.53 8.97310238.45310232.44
3934 3934J. Acoust. Soc. Am., Vol. 100, No. 6, December 1996 Hoffmeister
et al.
: Bioelasticity of tendon and myocardium
is interesting to note that the minima do not occur perpen-
dicular to the fiber axis. Rather, four equal minima occur at
56°, 124°, 236°, and 304°. The physical significance of this
feature is unclear, although we have obtained qualitatively
similar results using published values17 for the elastic stiff-
ness coefficients of a number of hexagonal crystals including
cadmium sulfide.16 The elastic properties of hexagonal crys-
tals are described by the same form of the elastic stiffness
matrix given in Eq. ~1!for a unidirectional fiber reinforced
material. Figure 1 also shows that Young’s modulus is larger
at all angles for tendon, and that tendon demonstrates a con-
siderably more pronounced anisotropy in Young’s modulus.
Table II provides a quantitative comparison of the two tis-
sues.
II. SENSITIVITY OF THE ANGULAR DEPENDENCE OF
YOUNG’S MODULUS TO VARIATIONS IN THE
STIFFNESS COEFFICIENTS
To assess the sensitivity of our results for Young’s
modulus on variations in the measured values of the elastic
stiffness coefficients, we varied each of the five independent
coefficients individually by a percentage of its measured
value, and used the software described in Sec. I to recompute
the angular dependence of Young’s modulus in the meridian
plane. The coefficients were varied within the limits estab-
lished by the physical realizability of the elastic stiffness
matrix. In general, the conditions of physical realizability
require that the diagonal elements and the determinants of
the principal minors of the elastic stiffness matrix be greater
than zero.17 Table III gives specific conditions for a material
exhibiting a unidirectional arrangement of reinforcing fibers
with a random transverse distribution. Within these limits,
each coefficient was varied by the minimum amount required
to produce a visible change in the anisotropy of Young’s
modulus. Figures 2 through 4 illustrate the results of this
analysis for formalin fixed bovine Achilles tendon.
Figure 2 illustrates the influence of 10% changes in our
measured values for c11 and c33. These variations changed
the numerical values for Young’s modulus at all angles. The
largest changes occurred for angles parallel to the fiber axis.
Figure 3 illustrates the influence of 50% changes in c44, and
90% changes in c66. Variations in c66 produced changes in
Young’s modulus at all angles, with the largest changes oc-
curring for angles perpendicular to the fiber axis. Variations
in c44 produced changes in Young’s modulus for all angles
except those parallel and perpendicular to the fiber axis. This
feature may be understood from Eqs. ~2!and ~3!which show
that Young’s modulus does not depend on c44 at these
angles. Figure 4 illustrates the influence of 10% changes in
c13. Variations in this coefficient produced changes in
Young’s modulus at all angles, with the largest changes oc-
curring for angles parallel to the fiber axis.
We repeated this analysis for myocardium, and observed
that 1% variations in c11,c33, and c13 produced changes in
the anisotropic features of Young’s modulus that were quali-
tatively similar to the influence of 10% variations in these
coefficients for tendon. The enhanced sensitivity to varia-
tions in these coefficients likely resulted from the relatively
small percentage difference between c11 and c33 for myocar-
dium. We also observed that 20% variations in c44 and 40%
variations in c66 affected the angular dependence of Young’s
modulus in a manner similar to that illustrated in Fig. 3 for
tendon. The result that relatively large percentage variations
in c44 and c66 are required to produce visible changes in the
anisotropy of Young’s modulus for either tissue is fortunate
given the considerable technical difficulties associated with
measuring these two coefficients.4,18,19
FIG. 1. Anisotropy of Young’s modulus in the meridian plane of formalin
fixed specimens of bovine Achilles tendon ~upper panel!and normal human
myocardium ~lower panel!.
TABLE II. Numerical results for Young’s modulus parallel and perpendicu-
lar to the fiber axis of formalin fixed specimens of bovine Achilles tendon
and normal human myocardium.
Tissue Eparallel
~GPa!
Eperpendicular
~GPa!
Eparallel
minus
Eperpendicular
~GPa!
Eparallel
Eperpendicular
Formalin fixed bovine
Achilles tendon 1.37 0.0706 1.30 19.4
Formalin fixed normal
human myocardium 0.101 0.0311 0.0699 3.25
TABLE III. Conditions of physical realizability for the elastic stiffness co-
efficients of a material exhibiting a unidirectional arrangement of reinforc-
ing fibers with a random transverse distribution.
c11.c66.0c11c33.c13
2
c33.0 and c44.0c33(c112c66).c13
2
3935 3935J. Acoust. Soc. Am., Vol. 100, No. 6, December 1996 Hoffmeister
et al.
: Bioelasticity of tendon and myocardium
III. VISUALIZATION OF THE ANISOTROPY OF
YOUNG’S MODULUS IN TWO AND THREE
DIMENSIONS
Young’s modulus may be represented by a direction and
a magnitude because the stress and strain lie along the same
axis. Consequently, Young’s modulus may be interpreted as
a vector-like quantity. This interpretation permits the genera-
tion of two-dimensional contours that display the modulus
for all directions in a given plane. Figure 5 illustrates our
results for formalin fixed bovine Achilles tendon in the me-
ridian plane ~upper panel!and transverse plane ~lower panel!
of the tissue. Note that these results are plotted on different
scales. Figure 6 illustrates our results for formalin fixed nor-
mal human myocardium in a similar manner.
Both tissues exhibit a substantial anisotropy in the me-
ridian plane, but are completely isotropic in the transverse
plane. This is because the fibers are assumed to be distrib-
uted randomly in the transverse plane. The transverse isot-
ropy of these tissues permits us to generate three-
dimensional surfaces representing Young’s modulus for all
spatial directions. This is accomplished by sweeping the two-
dimensional contours illustrated in the upper panels of Figs.
5 and 6 about the fiber axis. These surfaces may also be
generated directly from the elastic stiffness coefficients using
the software described in Sec. I. Figure 7 illustrates the sur-
faces we obtain for the formalin fixed specimens of tendon
and myocardium considered in this study. These surfaces
may be interpreted by imagining an observer located at the
origin inside one of these surfaces. The vector defined by
where the observer’s line of sight intercepts the surface de-
fines the magnitude of Young’s modulus in that direction. It
is important to note that the surfaces shown in Fig. 7 are
drawn to different scales. On the same scale, the long axis of
the surface representing our results for tendon would be
more than 13 times greater than that for myocardium.
IV. MODEL FOR THE FIBER ARCHITECTURE OF THE
HEART WALL
So far we have only considered the linear elastic prop-
erties of unidirectional soft tissues. In the specific case of
myocardium, we were careful to avoid complications arising
FIG. 2. Influence of 10% variations in the elastic stiffness coefficients c11
and c33 on the angular dependence of Young’s modulus for formalin fixed
tendon.
FIG. 3. Influence of 50% variations in c44 and 90% variations in c66 on the
angular dependence of Young’s modulus for formalin fixed tendon.
FIG. 4. Influence of 10% variations in c13 on the angular dependence of
Young’s modulus for formalin fixed tendon.
3936 3936J. Acoust. Soc. Am., Vol. 100, No. 6, December 1996 Hoffmeister
et al.
: Bioelasticity of tendon and myocardium
from the progressive shift in fiber orientation across the heart
wall by ultrasonically interrogating selected thin regions of
tissue exhibiting a unidirectional arrangement of
myofibers.1,4 We conclude by introducing a simple model
that we developed to extend our results for the linear elastic
properties of unidirectional specimens of soft tissues to sys-
tems exhibiting more complicated fiber architectures. As a
specific application, we use this model to estimate the angu-
lar dependence of Young’s modulus for the entire thickness
of the left ventricular wall.
As described in the introduction, the left ventricular wall
of the human heart is composed of layers of aligned myofi-
bers whose orientation depends on location within the heart
wall. Previous histologic and ultrasonic analyses have deter-
mined that normal myocardium exhibits a total transmural
shift in fiber orientation of approximately 120° from epicar-
dium to endocardium at a rate of approximately 10° per
millimeter.5,20 We model the fiber architecture of the left
ventricular wall by a stack of 121 elementary layers of tissue
arranged in 1° increments as illustrated in Fig. 8. Each el-
ementary layer represents a 100-
m
m-thick sheet of myocar-
dium containing a random transverse distribution of aligned
myofibers. Roman numerals are used to indicate the axes of
the coordinate system with the III axis corresponding to the
stacking axis, and the other two axes oriented such that the
fibers span a range of angles from 260° to 160° relative to
the I axis.
A number of techniques have been devised to evaluate
the elastic properties of man-made fiber reinforced stacked
composites. For certain high symmetry conditions, closed
FIG. 5. Two-dimensional representation of the angular dependence of
Young’s modulus for formalin fixed bovine Achilles tendon in the meridian
plane ~upper panel!and transverse plane ~lower panel!.
FIG. 6. Two-dimensional representation of the angular dependence of
Young’s modulus for formalin fixed normal human myocardium in the me-
ridian plane ~upper panel!and transverse plane ~lower panel!.
FIG. 7. Three-dimensional representation of the anisotropy of Young’s
modulus for formalin fixed bovine Achilles tendon ~upper panel!and normal
human myocardium ~lower panel!. Note that these results are drawn to dif-
ferent scales.
FIG. 8. Model for the fiber architecture of the left ventricular wall of the
human heart. The progressive transmural fiber shift is reconstructed by
stacking 121 thin unidirectional layers of myocardium in 1° increments.
3937 3937J. Acoust. Soc. Am., Vol. 100, No. 6, December 1996 Hoffmeister
et al.
: Bioelasticity of tendon and myocardium
form expressions may be obtained for the elastic stiffness
coefficients of these systems. One example is a biaxial com-
posite in which unidirectional layers are stacked with the
fibers alternately oriented at 0° and 90°.21 For our applica-
tion, however, no such symmetry exists. Our model therefore
follows an approach based on the law of mixtures that uses
the results we obtained for a unidirectional layer of myocar-
dium to predict the mechanical properties of the entire stack.
This is accomplished by specifying a stress axis in the I–II
plane, and averaging the contribution of each elementary
layer to obtain a mean value for Young’s modulus in that
direction. Based on previous investigations by others22–24 we
consider the two limits
E
u
51
121 (
f
5
u
260
u
160
E
f
,~4!
and
1
E
u
51
121 (
f
5
u
260
u
160 1
E
f
,~5!
where
u
represents the angle of the stress axis relative to the
I axis, and
f
represents the angle of the stress axis relative to
the fiber axis of an elementary layer.
Equations ~4!and ~5!are sometimes referred to as the
Voigt and Reuss bounds of the law of mixtures, and are
analogous to determining the spring constant of a system of
springs attached in parallel and series, respectively. These
two approaches may be considered to represent different lim-
its of the degree of interaction between the elastic elements
of the heart wall as defined by the elementary layers. If one
imagines each layer of the stack to represent a spring whose
spring constant depends on the angle of the stress axis rela-
tive to the fiber axis of the layer, then Eq. ~4!assumes that
the individual layers of the stack do not directly interact with
each other. Equation ~5!, on the other hand, assumes that the
elementary layers are directly coupled to one another in a
manner similar to springs attached end to end. In general,
Young’s modulus of the heart wall is expected to lie some-
where between these limits.25
To obtain numerical values of Young’s modulus from
our model, we developed a program written in the ‘‘C’’ pro-
gramming language for the Macintosh computer which ana-
lyzed the data presented in the lower panel of Fig. 1 to com-
pute Young’s modulus of the heart wall in 1° increments in
the I–II plane. Our results are illustrated in Fig. 9. The upper
panel of this figure displays Young’s modulus as a function
of angle relative to the I axis of the coordinate system as
computed using Eq. ~4!, and the lower panel displays the
result we obtained using Eq. ~5!. These two approaches pre-
dict only moderately different results for the angular depen-
dence of Young’s modulus. Equation ~4!yields a larger
value for Young’s modulus at all angles. However, the
maxima and minima in Young’s modulus occur at identical
angles for either approach. Maxima occur at 22°, 158°, 202°,
and 338°, and minima occur at 90° and 270°. The largest
difference between procedures in the predicted magnitudes
of Young’s modulus occurs at 0° and 180°, and the smallest
difference at 90° and 270°.
The results of our model satisfy certain intuitive features
regarding the mechanical properties of the left ventricular
wall. For example, our model predicts that the anisotropy of
Young’s modulus of the heart wall is considerably less pro-
nounced than the anisotropy of Young’s modulus of a unidi-
rectional specimen of myocardium. This feature reflects the
fact that, under normal physiologic conditions, the heart wall
must accommodate stresses that are not entirely unidirec-
tional. By orienting the myofibers over a distribution of
angles, the tissue may be better able to resist the approxi-
mately isotropic tensile forces produced in the left ventricu-
lar wall by the outward pressure of the blood filled cavity. It
is interesting to note that this function could also be accom-
plished by using a random orientation of myofibers. Conse-
quently, the progressive transmural shift in the orientation of
these fibers may contribute in other unspecified ways to the
function of the heart wall, and to its passive mechanical
properties.
V. DISCUSSION
Although the use of fresh as compared with fixed tissues
would be preferable for the determination of the anisotropy
of Young’s modulus, such tissues are difficult to handle in an
experimental situation. In particular, autolysis ~or enzymatic
degradation!ensues rapidly after excision unless the tissue is
fixed. However, the process of formalin fixation may also
change the mechanical properties of soft tissues. One indica-
tion of this change is that the tissues become noticeably
stiffer to the touch after fixation. The biochemical reactions
involved in the fixation of tissues with formalin are rather
complicated, but the main effect is to render proteins in-
soluble by binding together amino groups of lysine and
glutamine in adjacent protein chains. All of the ultrasonic
data used to analyze the anisotropy of Young’s modulus
were obtained from specimens of tissue that had been fixed
in a 10% buffered solution of formalin for a period of several
weeks to several months.
FIG. 9. Predictions for the angular dependence of Young’s modulus of a
specimen of the left ventricular wall possessing a 120° transmural shift in
fiber orientation.
3938 3938J. Acoust. Soc. Am., Vol. 100, No. 6, December 1996 Hoffmeister
et al.
: Bioelasticity of tendon and myocardium
A number of studies have been conducted to investigate
the effects of fixation on the ultrasonic properties of soft
tissues.26–28 For example, Shung and Reid reported that the
velocity of longitudinal mode ultrasonic waves propagated
perpendicular to the fibers of bovine myocardium increased
by approximately 1% after formalin fixation.29 If the density
of the tissue remains unaffected by formalin fixation, such a
change would produce a small increase in the elastic stiffness
coefficient c11. Similar increases may also be produced in
c33, but the effects on the remaining three coefficients, c13,
c44, and c66, are unknown.
Two recent studies suggest that the general principles of
the anisotropic elastic behavior elucidated in this report may
apply to measurements of fresh tissues. Recchia et al.
showed that values for anisotropy of ultrasonic integrated
backscatter in hearts of normal volunteer patients measured
by integrated backscatter imaging was approximately the
same as that measured in excised fixed animal hearts.30 Rec-
chia used integrated backscatter imaging and lateral gain
compensation in short axis images in vivo to quantify differ-
ences in scattering between the interventricular septal wall
~where myofibers are predominantly parallel to the ultrasonic
beam!and the posterior wall ~where myofibers are perpen-
dicular to the beam!. A 16-dB difference in backscatter was
observed between septum and posterior walls at the imaging
frequency of 2.5 MHz. Similar measurements were per-
formed in vitro in fixed canine and porcine hearts and the
magnitude of anisotropy was similar.
Rose et al. recently showed that backscatter from myo-
cardium depends on several factors including the elastic stiff-
ness constants, mass density, scatterer geometry, and ultra-
sonic frequency.31 The anisotropy of integrated backscatter
in formalin fixed excised human myocardium was well pre-
dicted with the use of the frequency domain Born approxi-
mation. Given the results reported by Recchia et al. which
showed that the magnitude of anisotropy was not signifi-
cantly different between fresh in vivo and fixed in vitro heart
tissues, the study by Rose et al. indicates that the set of pa-
rameters that determine ultrasonic scattering from myocar-
dium ~including the elastic stiffness coefficients!are unlikely
to be influenced by fixation to an extent that compromises
the measurement of anisotropy. Although absolute values of
these elastic stiffness constants probably change somewhat
after fixation, the general conclusions of this report regarding
the anisotropy of elastic behavior in soft tissue would appear
to be valid.
ACKNOWLEDGMENTS
We would like to thank Thierry D. Lhermitte, Ph. D., for
his helpful insights relating to this study. This work was
supported in part by NIH Grant Nos. HL 40302, HL 42950,
and an Established Investigator Award from the American
Heart Association ~SAW!.
1E. D. Verdonk, S. A. Wickline, and J. G. Miller, ‘‘Anisotropy of Ultra-
sonic Velocity and Elastic Properties in Normal Human Myocardium,’’ J.
Acoust. Soc. Am. 92, 3039–3050 ~1992!.
2B. K. Hoffmeister, E. D. Verdonk, S. A. Wickline, and J. G. Miller,
‘‘Effect of Collagen on the Anisotropy of Quasi-Longitudinal Mode Ul-
trasonic Velocity in Fibrous Soft Tissues: A Comparison of Fixed Tendon
and Fixed Myocardium,’’ J. Acoust. Soc. Am. 96, 1957–1964 ~1994!.
3B. K. Hoffmeister, S. M. Handley, E. D. Verdonk, S. A. Wickline, and J.
G. Miller, ‘‘Estimation of the Elastic Stiffness Coefficient c13 of Fixed
Tendon and Fixed Myocardium,’’ J. Acoust. Soc. Am. 97, 3171–3176
~1995!.
4B. K. Hoffmeister, S. E. Gehr, and J. G. Miller, ‘‘Anisotropy of the Trans-
verse Mode Ultrasonic Properties of Fixed Tendon and Fixed Myocar-
dium,’’ J. Acoust. Soc. Am. 99, 3826–3836 ~1996!.
5D. D. Streeter and W. T. Hanna, ‘‘Engineering Mechanics for Successive
States in Canine Left Ventricular Myocardium. II. Fiber Angle and Sar-
comere Length,’’ Circ. Res. 33, 656–664 ~1973!.
6E. Baer, J. J. Cassidy, and A. Hiltner, in Collagen, edited by Marcel E.
Nimni ~CRC, Boca Raton, FL, 1988!, Vol. 2, pp. 178–199.
7W. D. O’Brien, in Acoustical Holography, edited by L. Kessler ~Plenum,
New York, 1976!, Vol. 7, pp. 37–50.
8S. A. Goss, L. A. Frizzell, F. Dunn, and K. A. Dines, ‘‘Dependence of the
Ultrasonic Properties of Biological Tissue on Constituent Proteins,’’ J.
Acoust. Soc. Am. 67, 1041–1044 ~1980!.
9D. Amiel and J. B. Kleiner, in Collagen, edited by Marcel E. Nimni ~CRC,
Boca Raton, FL, 1988!, Vol. 3, pp. 223–251.
10 R. M. Hoyt, D. J. Skorton, S. M. Collins, and H. E. Melton, ‘‘Ultrasonic
Backscatter and Collagen in Normal Ventricular Myocardium,’’ Circula-
tion 69, 775–782 ~1984!.
11 K. T. Weber, ‘‘Cardiac Interstitium in Health and Disease: The Fibrillar
Collagen Network,’’ J. Am. Coll. Cardiol. 13, 1637–1652 ~1989!.
12 K. T. Weber and C. G. Weber, ‘‘Pathological Hypertrophy and Cardiac
Interstitium. Fibrosis and Renin–Angiotensin–Aldosterone System,’’ Cir-
culation 83, 1849–1865 ~1991!.
13 S. A. Wickline, E. D. Verdonk, A. K. Wong, R. K. Shepard, and J. G.
Miller, ‘‘Structural Remodeling of Human Myocardial Tissue After Inf-
arction,’’ Circulation 85, 259–268 ~1992!.
14 W. L. Bond, ‘‘The Mathematics of the Physical Properties of Crystals, ’’
Bell Syst. Tech. J. 22, 1–72 ~1943!.
15 S. M. Handley, J. G. Miller, A. E. Barnes, and E. I. Madaras, ‘‘Visualiza-
tion of the Engineering Moduli of Composites Using Measured Ultrasonic
Velocities,’’ Proc. IEEE Ultrason. Symp. 90CH2938-9, 1061–1064
~1990!.
16 S. M. Handley, ‘‘Physical Principles Pertaining to Ultrasonic and Me-
chanical Properties of Anisotropic Media and Their Application to Non-
destructive Evaluation of Fiber-Reinforced Composite Materials,’’ Ph.D.
thesis, Washington University, 1992.
17 B. A. Auld, Acoustic Fields and Waves in Solids ~Krieger, Malabar,
1990!, 2nd ed., Vol. 1.
18 L. A. Frizzell and E. L. Carstensen, ‘‘Shear Properties of Mammalian
Tissues at Low Megahertz Frequencies,’’ J. Acoust. Soc. Am. 60, 1409–
1411 ~1977!.
19 E. L. Madsen, H. J. Sathoff, and J. A. Zagzebski, ‘‘Ultrasonic Shear Wave
Properties of Soft Tissues and Tissuelike Materials,’’ J. Acoust. Soc. Am.
74, 1346–1355 ~1983!.
20 S. A. Wickline, E. D. Verdonk, and J. G. Miller, ‘‘Three-Dimensional
Characterization of Human Ventricular Myofiber Architecture By Ultra-
sonic Backscatter,’’ J. Clin. Invest. 88, 438–446 ~1991!.
21 T. D. Lhermitte, ‘‘Anisotropy of the Elastic Properties of Carbon/Epoxy
Composites—Study of the Ultrasonic Propagation, Dispersion and Back-
scattering,’’ Ph.D. thesis, Pierre & Marie Curie University, 1991.
22 O. L. Anderson, in Physical Acoustics, edited by W. P. Mason ~Academic,
New York, 1965!, Vol. IIIb, pp. 43–95.
23 S. Lees, R. S. Gilmore, and P. R. Kranz, ‘‘Acoustic Properties of
Tungsten-Vinyl Composites,’’ IEEE Trans. Sonics Ultrason. SU-20, 1–2
~1973!.
24 S. Lees and C. L. Davidson, ‘‘Ultrasonic Measurement of Some Mineral
Filled Plastics,’’ IEEE Trans. Sonics Ultrason. SU-24, 222–225 ~1977!.
25 Z. Hashin and S. Shtrikman, ‘‘A Variational Approach to the Theory of
the Elastic Behavior of Multiphasic Materials,’’ J. Mech. Phys. Solids 11,
127–140 ~1963!.
26 J. C. Bamber, C. R. Hill, J. A. King, and F. Dunn, ‘‘Ultrasonic Propaga-
tion Through Fixed and Unfixed Tissues,’’ Ultrasound Med. Biol. 5, 159–
165 ~1979!.
27 D. K. Nassiri, D. Nicholas, and C. R. Hill, ‘‘Attenuation of Ultrasound in
Skeletal Muscle,’’ Ultrasonics 17, 230–232 ~1979!.
28 A. F. W. van der Steen, M. H. M. Cuypers, J. M. Thijssen, and P. C. M.
3939 3939J. Acoust. Soc. Am., Vol. 100, No. 6, December 1996 Hoffmeister
et al.
: Bioelasticity of tendon and myocardium
deWilde, ‘‘Influence of Histochemical Preparation on Acoustic Param-
eters of Liver Tissue: A 5-MHz Study,’’ Ultrasound Med. Biol. 17, 879–
891 ~1991!.
29 K. K. Shung and J. M. Reid, ‘‘Ultrasonic Scattering From Tissue,’’ Proc.
IEEE Ultrason. Symp. 77 CH 1264-1SU, 230–233 ~1977!.
30 D. Recchia, J. G. Miller, and S. A. Wickline, ‘‘Quantification of Ultra-
sonic Anisotropy in Normal Myocardium with Lateral Gain Compensation
of Two-Dimensional Integrated Backscatter Images,’’ Ultrasound Med.
Biol. 19, 497–505 ~1993!.
31 J. H. Rose, M. R. Kaufmann, S. A. Wickline, C. S. Hall, and J. G. Miller,
‘‘A Proposed Microscopic Elastic Wave Theory for Ultrasonic Backscat-
ter from Myocardial Tissue,’’ J. Acoust. Soc. Am. 97, 656–668 ~1995!.
3940 3940J. Acoust. Soc. Am., Vol. 100, No. 6, December 1996 Hoffmeister
et al.
: Bioelasticity of tendon and myocardium
