Vol. 87, Issue 1, 211-221, July 1999
Wall thickness referenced to myocardial volume: a new
noninvasive framework for cardiac mechanics
Stewart
Denslow,
Seshadri
Balaji, and
Kenneth W.
Hewett
The Children's Heart Center of South Carolina, Medical University
of South Carolina, Charleston, South Carolina 29425
 |
ABSTRACT |
Dimensional variables measured for study of left
ventricular mechanics are subject to errors arising from difficulty in
determining zero-stress dimensions for use as a reference.
Based on a method validated for measurements within individuals, we
have devised an approach that facilitates comparison between
individuals while minimizing random scatter. We define an exact
mathematical index of strain,
ln(h0/h),
using wall thickness (h) referenced
to extrapolated wall thickness at zero-luminal volume
(h0).
Noninvasive data from rabbits, pigs, and humans all yielded highly
similar myocardial stress,
ln(h0/h),
and work values. The
stress-ln(h0/h)
relationship during afterload variation was constant among individual
pigs with a twofold variation in ventricular mass.
Stress-ln(h0/h) data from our analysis displayed lower scatter than either
pressure-volume data normalized to myocardial mass or
stress-ln(h0/h)
data referenced to end-diastolic dimensions. A Frank-Starling-like
curve with high correlation
(r2 = 0.96) was
constructed from single points from different pigs, suggesting a low
level of size and intersubject scatter. This method offers high
precision for noninvasive characterization of ventricular and
myocardial mechanics and for comparisons between subjects and between species.
ventricular function; wall stress; ventricular geometry
| |
INTRODUCTION |
RECENT REPORTS HAVE DESCRIBED a framework for
noninvasive analysis of myocardial behavior based on mean wall stress
and a strainlike quantity, the natural logarithm of reciprocal of wall thickness, ln(1/h) (21-23).
Wall thickness is used as a precise index of myocardial area under the
assumption of the approximate constancy of myocardial volume. This
approach has been shown to provide accurate estimates of myocardial
work per unit volume of tissue (21, 22). Within
individuals, it is sensitive to the effects of 
-blockers and other
inotropes (21). It has the further advantage of being based on easily
obtainable echocardiographic measurements.
The above investigations avoided the inclusion of a reference value for
wall thickness. When stress-strain analysis is used to investigate
constitutive relationships, such a reference measurement is generally
made at a point of zero net stress, a point not obtainable under
clinical conditions. Thus all analysis in these studies was in terms of
increments in the variable ln(1/h)
that indexed strain.
The lack of a reference, although practical for determining energetic
and geometric changes, prevents direct comparison of values between
different individuals. This lack also causes the units of the
strainlike variable to be ambiguous. We investigated whether use of a
calculated reference would provide a basis for comparison of values
between individuals, as well as avoided the use of the ambiguous units
of natural logarithm of reciprocal distance. Because myocardial mass
has been shown to be effective in normalizing other descriptors of
myocardial behavior (4, 15), we specifically examined references
calculated as cube-root functions of this quantity to produce a value
with units of length rather than volume. The present paper discusses
results obtained by using a reference that is the extrapolated value of
wall thickness that would occur if the luminal volume could go to zero.
Glossary
| A |
Area of a truncated conical section of myocardium,
cm2
|
| D |
Instantaneous endocardial short-axis of left ventricle, cm
|
| Dm |
Instantaneous midwall short-axis of left ventricle, cm
|
| Dm0 |
Midwall short-axis of left ventricle at zero-luminal volume, cm
|
| ESPVR |
End-systolic pressure-volume ratio
|
| FS |
Fractional shortening of the left ventricle
|
| h |
Instantaneous measured wall thickness, cm
|
| h0 |
Wall thickness at zero-luminal volume, cm
|
| L |
Endocardial long-axis of left ventricle, cm; mean sarcomere length,
µm
|
| LV |
Left ventricle
|
| L0 |
Mean sarcomere length at zero-luminal volume, µm
|
| P |
Blood pressure in ventricular lumen,
g/cm2
|
| p, q,
r, a,
b |
Arbitrary intermediate quantities for solving a cubic equation
|
| T |
Mean of circumferential and meridional tension, g/cm
|
| V |
Volume of a truncated conical section of myocardium,
cm3
|
| Vc |
Cavity volume of the left ventricle,
cm3
|
| Vlum |
Left ventricular luminal volume,
cm3
|
| Vmyo |
Left ventricular myocardial volume,
cm3
|
| VT |
Left ventricular total volume (luminal + myocardial volume),
cm3
|
| Vw |
Left ventricular wall volume, cm3
|
| V0 |
Extrapolated left ventricular luminal volume at zero pressure,
cm3
|
| VCF |
Velocity of circumferential fiber shortening, circumferences/s
|
| VCFc |
Corrected velocity of circumferential fiber shortening,
circumferences/s
|
| W |
Work done by a truncated conical section of myocardium,
g · cm
|
 |
Arbitrary proportion of myocardium
|
 |
Mean of circumferential and meridional wall stress or fiber stress,
g/cm2
|
| |
THEORY |
Definition of variables.
Two widely used descriptors of material deformation are LaGrangian
strain, (x 
x0)/x0,
and natural strain,
ln(x/x0).
The natural strainlike quantity used by Nakano et al. (21-23) is
not referenced and thus is not directly comparable in magnitude between different hearts. One logical reference dimension,
h0, would be the
wall thickness at zero external load, producing the variable ln(h0/h).
However, the roughly tangential approach of the myocardial stress-strain relationship to the zero external stress axis (10, 39)
results in large random errors in any measurement of this reference
point. Consequently, we sought a reference that was more precisely
determinable despite the fact that it would not correspond to zero net stress.
The net effect of a reference thickness taken at a new point is to
cause a constant additive shift in all
ln(h0/h)
values parallel to the
ln(h0/h)
axis. This is due to the identity
ln(h0/h) = ln(h0) ln(h). A new
h0 is simply a
different additive constant that does not change the shape of empirical
stress-ln(h0/h) contours.
Because of the demonstrated usefulness of myocardial mass (and the
directly proportional myocardial volume) in normalizing descriptors of
myocardial mechanics (4, 15), we sought a reference value directly
related to this quantity. The line plotted in Fig.
1 is the relationship between wall
thickness and luminal dimension based on the approach of Hugenholtz et
al. (17), which models the myocardial wall as an ellipsoid of constant
wall volume. This assumption has been shown to provide an accurate
prediction of wall thickness over the physiological range of
ventricular contraction (17). Extrapolation beyond the physiological
range to zero-luminal diameter provides a clearly defined calculated value that is precisely representative of the myocardial volume while
having the proper units (length) for use as a reference quantity.
Although this value is not obtainable under physiological conditions,
we felt that its clear definition and precise relationship to
myocardial mass strongly suggested its appropriateness. This approach
is also analogous to the use of allometric equations in which reference
variables (e.g., height, weight, or body surface area) are taken to a
whole or fractional power so as to provide more appropriate dimensional
units (11, 16).
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Fig. 1.
Relationship of wall thickness to luminal diameter for thick-walled
ellipsoid of incompressible myocardium. This is the relationship used
by Hugenholtz et al. (17) for determining wall thickness at any point
in cardiac cycle based on end-diastolic measurements.
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Tangential wall-stress relation to radial thickness.
Description of the mathematical relationship between (radial) wall
thickness and (tangential) wall stress has been published previously by
others (21, 22). Briefly, approximation of the myocardium as
incompressible allows substitution of a variable (thickness) that lies
along the transmural dimension for the mean of the dimensional
variables (arc lengths or areas) that lie in the plane of the wall.
Contraction of the myocardium is described in terms of perimeter
tension and area strain of a truncated conical region of the
ventricular wall. Work, W, done by this region is the integral of
perimeter tension, T, with respect to change in area,
A: W = 
TdA. Dividing by
volume, V, of the wall region gives work per unit volume of myocardium: W = (T/V)dA. The
volume of the region is the product of the area and the thickness,
h; hence the expression can be written
as W = (T/h)(dA/A).
Tension over thickness is the mean wall stress, , from equatorial
and meridional directions, whereas
dA/A
is equal to d(lnA). The formula becomes W = d(lnA).
Area, A, can be replaced by
V/h, where V is again the constant
volume of the truncated conical region. Because V is constant, the
differential can be manipulated for its elimination:
d[ln(V/h)] = d[lnV + ln(1/h)] = dln(1/h). The result is W = d[ln(1/h)].
For the present extension of this approach, we took the integration
between the calculated reference of extrapolated thickness at
zero-luminal volume,
h0, and
instantaneous thickness, h, measured from M-mode echocardiograms. The general result is W = ln(h0/h), where
ln(h0/h)
is an exact mathematical index of natural strain in the circumferential
direction, ln(length/reference length). In that their product is work,
and
ln(h0/h)
form a pair of intensive and extensive mechanical variables analogous
to pressure and volume or force and length. However, the work expressed
is per unit volume of material rather than total work.
The reference value,
h0, is in the
numerator rather than the denominator. When the index of strain is
defined this way, its value increases in parallel with luminal diameter
and volume, i.e., as the wall becomes thinner. Work calculated by using
a more conventional definition,
ln(h/h0),
would have a minus sign during ejection of blood. We chose to avoid
this potential source of confusion by keeping the inverted form
resulting from the above derivation. The zero point of this index of
strain corresponds to the zero value of luminal volume, resulting in a
qualitative similarity between pressure-volume plots and
stress-ln(h0/h) plots.
Previous work (21) has demonstrated shifts in data within individuals
in response to -blockers and other inotropes. The addition of a
reference dimension does not alter this sensitivity. Because all data
from one individual will be referenced to the same value based on the
myocardial mass
(h0), all data
will be shifted identical distances with the result that prereference shifts will remain. Relative position is only changed between data
points obtained from hearts of different mass.
Recent reports provide some separate support for the usefulness of
referencing to zero-volume dimensions (2, 3, 10). These studies found
that, during ejection, the instantaneous mean sarcomere length,
L, is accurately predicted by a
function including calculated sarcomere length at zero-luminal volume,
L0, and the ratio
of luminal and wall volumes, Vlum
and Vw:
L = L0 [1 + (3Vlum)/Vw]1/3.
Thus the sarcomere length can be referenced to zero-volume length in a
useful predictive equation:
L/L0 = [1 + (3Vlum)/Vw]1/3.
| |
METHODS |
Human studies.
Patient data were collected retrospectively from records of the
Division of Pediatric Cardiology at our institution. Data were selected
from patients who underwent 1)
successful ablation of substrate of intermittent supraventricular
tachycardia (Wolff-Parkinson-White or atrioventricular node reentrant
tachycardia) in a heart without shunts, incompetent or stenosed valves,
or restrictions to flow in the great vessels; and
2) same- or next-day
postcatheterization echocardiographic study. M-mode echocardiographic
measurements were taken perpendicular to the long axis of the ventricle
at the tops of the papillary muscles with the cursor directed through the septum to a point between the papillary muscles. Measurements taken
from the M-mode recordings were end-diastolic wall thickness (mean of
posterior wall and intraventricular septum), end-diastolic luminal
diameter, and end-systolic luminal diameter. Because blood pressures
were not available as part of the echocardiographic studies, systolic
and diastolic cuff pressures recorded during catheterization were used.
Sixteen patients were included (ages 6-32 yr, mean 14 yr).
Calculated ventricular masses (including intraventricular septum; see
Calculations below) ranged from 37 to
124 g.
Animal models.
All animals were treated and cared for in accordance with the National
Institutes of Health Guide for the Care and Use of Laboratory Animals (National Research Council,
Washington, DC, 1996).
Swine studies.
Ten Yorkshire pigs (25-30 kg) of either sex were part of studies
of congestive heart failure in which animals were subjected to rapid
atrial pacing. For the present study, data were collected from animals
that had yet to receive pacing, after 2 wk of recovery from surgery.
The surgical techniques of pacemaker and aortic access port
implantation have been described previously (32, 34).
Pigs were presedated with diazepam (10 mg) and brought to the
laboratory in a sling. For recording of aortic pressure tracings, a
vascular access port was entered by using a 12-gauge Huber needle attached to a saline-filled tube and a Statham 23ID (Gould, Oxnard, CA)
pressure transducer. Further sedation was achieved with 5- to 10-mg
boluses of midazolam given through the vascular access port. Surface
electrocardiographic (ECG) electrodes were attached to each limb, and
standard ECG leads I, II, and III were displayed on a physiological
recorder (EVR-16, Electronics for Medicine, Overland Park, KS).
Two-dimensional and M-mode echocardiographic studies of left ventricle
(LV) function were performed (2.25-MHz transducer, ATL Ultramark VI,
Bothell, WA) from the right parasternal approach. LV dimensions were
measured at end systole and end diastole as described for patients.
Calculated LV myocardial masses (see Calculations below) ranged from 23 to
64 g.
Six pigs were given variable doses of phenylephrine to increase
afterload. Phenylephrine infusion started at a rate of 1.3 µg · kg
1 · min1
and increased in such a manner so as to obtain four to seven different
simultaneous M-mode tracings and blood pressures. This allowed
determination of four to seven different paired values of afterload and
fractional shortening for each pig.
Rabbit studies.
Data were obtained from three sham-operated rabbits (New Zealand White,
3.5-4.5 kg, 5 mo of age) that served as controls in a study of
pacing-induced congestive heart failure. Procedures for instrumentation
in this model have been published previously (33). The animals were
administered 4 mg/kg sc of enrofloxacin (Baytril, Miles, Shawnee
Mission, KS) twice per day on postoperative days
1 and 2. In 4 wk of
time, two-dimensional and M-mode echocardiography (5-MHz transducer,
ATL Ultramark VI) were used to image the LV from the right parasternal
approach (33). The same ventricular dimension measurements were taken
in rabbits as in pigs and patients. Before being brought to the
laboratory, rabbits were sedated with 10 mg of intramuscular midazolam
(Versed, Hoffman-LaRoche, Nutley, NJ) and placed in a custom sling, and
an ECG was established.
Pressure was obtained at the time of the terminal experiment with a
fluid-filled catheter advanced into the LV via the right carotid
artery. The catheter was connected to a previously balanced and
calibrated Statham 23ID pressure transducer (Gould). LV masses obtained
at autopsy ranged from 3.4 to 4.7 g wet wt.
Calculations.
For determination of wall thickness at other than end diastole, we used
the approach validated by Hugenholtz et al. (17) with luminal volume
values determined as in Zile et al. (40). The latter method assumes
constant myocardial volume and is based on the modeling of the LV as a
half-ellipsoid of circular cross section with a constant ratio of
epicardial long and short axes of 1.2 (40). Wall thickness at the apex
is assumed to be 0.4 times that at the base (40). Accordingly, using
1) averaged end-diastolic wall
thickness, 2) end-diastolic luminal
dimension, and 3) end-systolic
luminal dimension, we calculated end-systolic (end-ejection) and
zero-luminal wall thicknesses (Eqs. A5 and A6, respectively, in the
APPENDIX). Start-ejection wall
thickness was taken as equal to end-diastolic thickness.
Wall stresses (mean fiber stress) at start and end ejection were
calculated based on end-diastolic and end-systolic arterial pressures,
respectively, with the use of the formula of Regen (25). End-systolic
pressure was estimated as equal to the calculated mean aortic pressure
(28, 29)
Mean
fiber stress, in g/cm2, was
calculated as
where
total volume is the sum of the luminal volume and myocardial volume
(25). Mean fiber stress is the mean of orthogonal fiber stresses
(meridional and circumferential) on an element of myocardium (25).
LV luminal and myocardial volumes, including free wall and septum, were
calculated based on the model of Hugenholtz et al. (17). We used the
Zile geometric model (40) for the determination of the long axis of the
ventricle. We recognize that use of this model may have resulted in a
bias in determined volume and mass values. However, because we were
concerned only with the relative sizes of the ventricles, the use of a
consistent formula was felt to be a valid approach. Myocardial mass was
calculated as myocardial volume times 1.05.
The area under ejection trajectories was calculated as mean stress
times the difference in
ln(h0/h)
values at start and end ejection. The mean stress was calculated as the
simple mean of the start-ejection and end-ejection stresses.
Ejection time was measured from M-mode recordings as the time between
start of contraction and end of contraction.
Data are presented as means ± SD. Statistical difference between
regression lines was tested by using confidence ellipses (24). A
Bonferroni correction for multiple tests was used in tests for
significance by analysis of variance (35). Spearman's rank correlation
was used to test for ordering within clusters (31).
| |
RESULTS |
Size independence of stress,
ln(h0/h),
and work variables.
Figure 2 shows a plot of mean values and
95% confidence intervals of stress and
ln(h0/h)
at beginning and end systole obtained from rabbits, pigs, and patients.
Estimated ventricular masses had means (and ranges) of 4 (3-5), 40 (23-64), and 78 (37-124) g for rabbits, pigs, and humans,
respectively. On the basis of the form of the Laplace law, the
similarity of the stress values might be expected, both within and
between species, due to the similarities in cardiac anatomy and blood
pressures (102/76, 114/80, and 109/62 mmHg for rabbits, pigs, and
humans, respectively). Again, because of the form of the Laplace law,
one may expect that an index of strain would vary only a small amount
because the proportional anatomy of these ventricles is similar both
within and between species. Only differences in inotropic state would be expected to result in variations of
ln(h0/h)
as it does for ln(1/h). The plot
indeed displays the independence from ventricular size of not only
stress but also
ln(h0/h).
The variation among species is on the same order as that within species
as indicated by the error bars. Multiple one-way analysis of variance
of stress and
ln(h0/h)
values at start and end ejection detected no statistically significant
differences among these points.
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Fig. 2.
Mean wall stress and
ln(h0/h)
at start and end ejection for 3 species. No. of subjects is in
parentheses. Mean masses were 4, 40, and 78 g for rabbits, pigs, and
humans, respectively. Error bars represent 95% confidence intervals.
Values from all sources are similar despite 20-fold variation in mean
ventricular mass.
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Figure 3 shows the same plot as in Fig. 2
with the addition of
stress-ln(h0/h)
values that were calculated from measurements extracted from cardiac
literature. The species studied were rats (18, 19), lambs and sheep
(1), and humans (9). These results, although based on data from
completely independent studies, agree closely with present results.
This pattern suggests that these quantities are insensitive to
interobserver variation, and that interspecific differences in these
quantities, if present, are small for these species. End-systolic
values, which would be expected to vary less with likely differences in
loading between experimental populations, show closer similarity.
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Fig. 3.
Comparison of present
stress-ln(h0/h)
results with results derived from published ventricular dimensions and
pressures. No. of subjects is in parentheses. Values from disparate
sources are similar to values from present study.
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By using start- and end-ejection values, ejection work was calculated
by approximating the ejection trajectory in the
stress-ln(h0/h) plane as a straight line. Validity of this approximation for normally ejecting ventricles in dogs and humans is supported by published studies of stress-strain loops (22, 37). On the basis of the Laplace
law relating stress to pressure and proportional geometry, the
similarity in ventricular geometry and pressure waveforms among the
species included in this study should result in similar ejection
trajectories. The area under this line segment represents the work per
unit myocardium during ejection. The values are similar, despite a
roughly 150-fold range in myocardial mass (Table
1). Observed variations are on the order of
what would be expected within single individuals because of preload and
afterload variations. The values are also similar to those reported by
Nakano et al. (Ref. 22; 40-96
g · cm1 · cm3)
for patients for the full cardiac cycle.
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Table 1.
Ejection work per unit volume of myocardium calculated from mean stress
and ln (h0/h) values at start- and
end-ejection
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The response of end-systolic stress and
ln(h0/h)
to afterload variation from graded doses of phenylephrine is shown in
Fig. 4. End-systolic points for each pig
fall along straight lines with little scatter
(r2 = 0.94-0.99). These lines cluster tightly without ordering according to the approximately twofold variation in ventricular mass (23-40 g, mean 31 g) [Spearman's rank correlation between mean
end-systolic ln(h0/h)
and myocardial mass not significantly different from zero]. None
of the regression lines shows a statistically significant difference
from the others.
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Fig. 4.
Relationships of end-systolic stress and
ln(h0/h)
during graded phenylephrine infusion in 6 different pigs. Data points
from each pig (from 4 to 7 points/animal) are denoted by a different
symbol. Solid lines are regression lines through each data set
(n = 4-7;
r2 = 0.94-0.99). Regression lines displayed no significant differences
or ordering according to mass.
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Figure 5 combines Fig. 4 with results
calculated from published dimension values from six open-chest dogs
undergoing afterload variation because of aortic constriction (23). The
values were obtained from plotted data with the use of a calibrated
digitizer tablet. Based on reverse calculation from values of derived
quantities in tables, the masses of these canine ventricles ranged from
19 to 67 g with a mean of 41 g. As an indicator of the location of the
data set from each dog, we calculated the mean
x-position within each animal's set
of points. In the original analysis, these mean
ln(h0/h)
values had a total range of 0.496. The range of the present results was
reduced by ~50% to 0.26 with a mean position very close to, and not
statistically different from, that of the swine results.
Changes in both mass and species resulted in little or no position
change in the
stressln(h0/h)
domain.
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Fig. 5.
Relationships of end-systolic stress and
ln(h0/h)
calculated from results obtained by Nakano et al. (23) during variable
occlusion of aorta in open-chest dogs. Data from each dog are denoted
by a different symbol. Solid lines are regressions through reanalyzed
data. Dotted lines are regressions from Fig. 4. There was no
significant difference detectable between the two sets of regression
lines (unpaired t-tests of slope and
intercept with Bonferroni correction).
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The values of both slope and intercept from the recalculated dog
results range from below to above those of the pig data. The variation
in parameter values is much greater in the recalculated dog data than
in the pig data. This variation might be due to a round-off error
introduced when published data were reanalyzed, or it may represent
real variations in the slopes and intercepts that are characteristic of
the species. As would be expected because of the complete overlap of
ranges, no statistically significant difference was found between the
population means of the canine and swine regression parameters (slope
and intercept) with the use of simultaneous
t-tests of slopes and intercepts with
Bonferroni correction.
Preliminary investigation of the
stress-ln(h0/h)
relationship under constant afterload was based on data from two
literature sources in which the end-systolic pressure-volume ratio
(ESPVR) (36) was investigated (15, 39) in isolated, cross-circulated hearts from rabbit (5 g) and dog (131 g), which contracted
isovolumically and displayed quite different ESPVRs. In contrast to the
afterload variation results discussed in Size
independence of stress,
ln(h0/h), and work
variables, these data display clearly nonlinear
ESPVRs. This is probably due to the lower range of
preloads that is attainable with this type of preparation. The higher
preloads observed under afterload variation apparently span only a
linear portion of the total curve.
As can be seen in Fig. 6, the
stress-ln(h0/h)
curves are almost superimposable, despite the species and size
differences. If the data are presented in the
ln(1/h) format of Nakano et al. (21-23), i.e., without referencing, the mean
x-positions of the data sets are
0.62 and 0.60. After referencing, the mean positions are 0.56 and 0.51.
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Fig. 6.
Reanalyzed data from rabbit ( ) and dog ( ) isolated,
cross-circulated hearts (15, 39). End-systolic data from isovolumic
contractions with varying preload were used to generate results shown
here. Calculated end-systolic
stress-ln(h0/h)
relationships are nearly superimposable.
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Precision of stress and
ln(h0/h)
variables.
As opposed to the direct measurement of volume carried out for
pressure-volume results (4, 15), noninvasive approaches require the
calculation of volumes from linear measurements. Such calculated
volumes will have a magnified level of scatter (lower statistical
precision or sensitivity) because of the cubing of measured values. To
demonstrate this effect, the
stress-ln(h0/h) values from one of the phenylephrine infusion experiments in Fig. 4 is
reproduced in Fig.
7A,
whereas Fig. 7B displays values of pressure and luminal volume over mass that have been calculated from
the same raw data. The value of the coefficient of determination, r2, is the
fraction of total variance in the data that is accounted for by the
regression relationship (24) and thus is a direct indicator of the
precision of the data. The
r2 value was
higher for the present analysis (0.95 vs. 0.72), indicating a lower
level of scatter. As shown in Table 2, this
pattern of r2
values was true for data from all six pigs from the
phenylephrine-infusion experiments (Fig. 4).
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Fig. 7.
Plots of alternate descriptors calculated from same data as those in
Fig. 4. Displayed data were taken from animal with greatest no. of
points recorded (n = 8) during graded
phenylephrine infusion. A: plot of
stress-ln(h0/h)
data calculated by presently proposed method.
B: plot of pressure vs. volume/mass
values calculated from same data as used for
A. C:
plot of stress vs.
ln(h0/h)
referenced to end-diastolic dimension calculated from same data as used
for A. Pattern of lower
r2 values for
alternate approaches was observed in all 6 pigs displayed in Fig. 4
(see Table 2).
hed and
hes,
End-diastolic and end-systolic wall thickness,
respectively.
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Table 2.
Values of slope, intercept, and r2 for regressions of three
pairs of variables calculated from raw data used to produce Fig. 4
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Other noninvasive methods of ventricular performance assessment, such
as FS (7), velocity of circumferential fiber shortening (VCF) (5), and
corrected VCF (VCFc) (8), use
end diastole as a reference state. This is also true for
some invasive methods (20). However, natural variation in end-diastolic
distension from beat to beat would be expected to result in additional
random scatter not accounted for by an underlying
stress-ln(h0/h)
relationship. Figure 7C displays
values of stress paired with
ln(h0/h),
calculated from the same raw data set as Fig.
7A but by using end-diastolic thickness as a reference rather than thickness at zero-luminal volume.
As expected, the value of
r2 was higher
(0.95 vs. 0.87) when the reference was the zero-luminal thickness.
Again, this pattern was true for all six pigs from the
phenylephrine-infusion experiments.
Thus in all six sets of regression results, the present approach
produced the highest
r2 value and
highest precision. With the use of a sign test (31), this result was
tested against a null hypothesis of the
r2 value of the
present approach not being the highest (i.e., one or both of the other
approaches having equal or higher
r2). The test
showed significance with a P value of
0.016.
Ejection energetics and end-diastolic
ln(h0/h).
If indeed the tested method has a high precision, it should allow the
discernment of underlying patterns that are obscured by random scatter
in other methods. Accordingly, a final area of investigation was the
relationship of ejection work per unit volume of myocardium to
end-diastolic
ln(h0/h).
This relationship is analogous to a Frank-Starling curve relating
stroke volume to end-diastolic luminal volume. Single-point data, each
based on M-mode and pressure measurements from a single time point, from 10 healthy control pigs of similar age are plotted in Fig. 8. Linear regression analysis demonstrated
a high correlation (r2 = 0.96)
between ejection work and end-diastolic
ln(h0/h).
The probability of this relationship of points occurring by chance, when no actual relationship exists, is <0.0001
(P value of the regression). As would
be expected based on the Frank-Starling relationship, work increased
with diastolic distension [high values of
ln(h0/h)].
Remarkably, the present result has been obtained by using independent
baseline measurements from 10 different animals. Although the
contractilities, and hence Frank-Starling curves, of these animals
might be expected to be similar, it would be predicted that
measurements from separate animals would be confounded by individual
variations in size and physiology, thus lowering correlation.
View larger version (15K):
[in this window]
[in a new window]
|
Fig. 8.
Plot of ejection work per unit volume of myocardium against
start-ejection
ln(h0/h).
Each point represents single baseline measurement on a separate pig
(n = 10). Solid line is regression
line through points, and dotted lines are two SEs above and below the
line. SEE, SE of estimate.
|
|
Regression with the use of end-systolic points, points not expected to
follow a Frank-Starling relationship, showed no correlation (r2 = 0.02, P < 0.71), suggesting that the
end-diastolic correlation was not due simply to correlation between
work and overall ventricular mass. Regression was also performed by
using data obtained from the same animals after 3 wk of rapid atrial
pacing used to induce heart failure. The heart-failure animals would be
expected to have a large range of (reduced) contractilities (32) and
thus would not be expected to share similar Frank-Starling
relationships. The regression failed to show any correlation
(r2 = 0.05, P < 0.56).
We also investigated the behavior of calculated ejection power
(ejection work/ejection time) with end-diastolic
ln(h0/h)
in the same set of control pigs (Fig. 9).
We again found a high correlation (r2 = 0.89) with
end-diastolic
ln(h0/h),
but not end-systolic
ln(h0/h) (r2 = 0.01, P < 0.78) or end-diastolic
ln(h0/h),
after induction of heart failure by rapid atrial pacing
(r2 = 0.01, P < 0.76).
View larger version (15K):
[in this window]
[in a new window]
|
Fig. 9.
Plot of ejection power per unit volume of myocardium against
start-ejection
ln(h0/h).
Each point represents a single baseline measurement in a separate pig
(n = 10). Solid line is regression
line through points, and dotted lines are two SEs above and below the
line.
|
|
| |
DISCUSSION |
The method of Nakano et al. (21-23) offers the advantages of being
noninvasive and physics based. It has been demonstrated to
provide accurate estimates of myocardial work and to be sensitive to
myocardial inotropic state. The present study has demonstrated the
additional precision achieved by referencing the index of strain,
ln(1/h) to calculated thickness at
zero-luminal volume, h0. Measured
values of stress,
ln(h0/h),
and work per unit volume of myocardium from different-sized individuals
of one species are very similar when zero-volume referencing is used.
Furthermore, the similarity of myocardial energetic and mechanical
properties among different mammalian species, widely demonstrated at
tissue and biochemical levels, can be directly observed in vivo by
using this noninvasive approach.
Despite the noninvasive nature of the method, patterns of stress and
ln(h0/h)
under varying afterload were found to have high correlations and
remarkable similarity among ventricles of differing size in pigs.
Position shifts in plots of stress vs. strain index, observed in
previous studies employing the unreferenced strain index
ln(1/h) (21), thus have significance
for absolute as well as relative value. The absolute location of a
stress-ln(h0/h) curve could potentially indicate an abnormal myocardial state without
reference to a control value within the same animal. Initially variant
or abnormal individuals could be identified and analyzed accordingly.
Results obtained from published data in a different species (dog),
collected completely independently from the present work, also
demonstrated the increased clustering of results within a species that
is achieved by the referencing to
h0. The mean
positions of the clustered data points were similar between the species as were the mean slopes and intercepts of regression lines. Again, this
allows analysis in terms of absolute as well as relative position of
the stress vs. strain index curves. Whereas the dog curves showed a
greater variation in parameters (mean abcissa, slope, and intercept),
the means of these parameters were not different from those of the
swine data. This difference in variance of the location parameter would
not be detectable under systems in which location is not adjusted
according to myocardial mass, such as when
ln(1/h) is used.
We have demonstrated the lower random scatter (higher precision) that
results from the present approach, compared with approaches based on
calculated volumes or on end-diastolic references. Random scatter was
also low enough to allow the construction of a Frank-Starling-like curve by using single points from different animals. The known underlying relationship was discernible despite variations between individuals.
Referencing to zero-volume thickness may seem problematic because such
dimensions are not physiological. This thickness, however, is a
characterization, or parameter, of the basic geometry of the ventricle
rather than a physiological measurement. It would be mathematically
equivalent to our present method to use a dimension occurring at a
different luminal volume, as long as that luminal volume were a
constant proportion of myocardial volume. For example, calculating a
reference thickness at luminal volume equal to myocardial volume,
rather than at zero lumen, would lower all presented
ln(h0/h) values by a constant value, 1.056. Whereas most values would shift to
negative, all correlations among the data and similarities between data
sets would remain unchanged. Although a choice among mathematically
equivalent reference points cannot avoid being somewhat arbitrary, we
feel that the choice of a point at zero volume avoids possible
confusion engendered by negative values of
ln(h0/h)
as well as resulting in the coincidence of zero
ln(h0/h) and zero-luminal volume.
Comparison to mass-based normalizations.
Normalization of ventricular measurements to the myocardial mass has
been used to standardize pressure-volume relations (4, 15). The present
method is analogous to such an approach because wall thickness at
zero-luminal volume is directly related to the total myocardial volume.
However, normalizations that are based on mass utilize a ratio
involving two different quantities (luminal volume and wall mass) in
contrast to the ratio of two values of the same quantity (wall
thickness) as in the present approach. The combination of values from
different geometric structures cannot be used as an index of strain in
analysis, thus complicating possible comparisons to stress-strain
behavior (curve shapes) in other situations, such as isolated
myocardial strips and trabeculae (6, 14).
Because, due to its noninvasiveness, the present method relies on
distance measurements rather than direct measurement of volume, only
calculated volumes can be obtained. The present approach avoids this
demonstrated source of scatter (Fig. 7) by measuring and calculating
linear quantities only.
Results comparable with the present study are theoretically obtainable
when midwall measurements are normalized or referenced to calculated
values at zero-luminal volume rather than end-diastolic dimensions, as
have been used previously (12, 13, 20, 27, 30). The mathematical
relationship between wall-thickness ratios and midwall-diameter ratios
is not linear, however (see the
APPENDIX); therefore, the shapes of
the relationships with stress, work, and power may also be different.
The use of wall thickness and luminal diameter in the present
investigation is based on the ease of obtaining these measurements
through echocardiographic imaging.
Physical meaning of variables.
The present study investigated stress,
ln(h0/h),
work, and power. These variables are fundamental physical quantities
expressed in fundamental physical units. Accordingly, these
noninvasively determined values may be directly compared with values
obtained from experiments that used other fundamental quantities, such as force and distance, pressure and volume, or generated heat. This
situation contrasts with that of other noninvasively obtained quantities, such as FS (7), VCF (5), and
VCFc (8), that can only be
compared with other measurements of the same indexes.
The work of Nakano et al. (21) using
ln(1/h) suggests that the
relationship of that variable to mean tangential stress can be an index
of contractility. Comparison of the present work with their results
would suggest that the
stress-ln(h0/h)
relationship could be used in a similar fashion. We did not explicitly
manipulate contractility, however, and, therefore, have not addressed
this question. Accordingly, our main conclusion is that the confounding effects of size and species on measurements of mechanical and energetic
variables appear to be substantially reduced or eliminated by our approach.
Limitations of the present work.
We chose to use a particular geometric model of the LV that, as with
any geometric model, contains some inaccuracy. The consistent use of
this model for all calculations should result in equal bias for all
results and thus allow the valid comparison of values.
Because it is a material property,
ln(h0/h)
should be only slightly affected by the shape of the chosen geometric
model. Furthermore, because the stress variable used here is the mean
of equatorial and meridional stress, changes in shape have little
effect on its final value (2). We tested these predictions and found <1% change in calculated values of stress or
ln(h0/h)
when the geometric model was varied in the ratio of the long axis to
the short axis from 1 to 1.4. Thus our particular choice of
long-axis/short-axis ratio should have a negligible effect on the
accuracy of the results.
There may be a varying degree of inaccuracy of the geometric model,
depending on the size and, especially, species origin of the heart.
This potential source of error cannot be eliminated from the present
study. Considering the similarity of values among the species
investigated in the present work, it would seem that this error is not
of great magnitude.
There may be a high percentage of error in measurements of small values
of wall thickness. Despite the fact that many of the data points were
single measurements of wall thickness, little scatter was observable in
plots of end-systolic stress and
ln(h0/h) under varying afterload. The high correlations obtained in these experiments would suggest that this source of measurement error had
little effect on the results.
The systolic trajectory in the
stress-ln(h0/h)
relationship was approximated as a straight line, as suggested for
results in dogs and humans. It is not known how consistently this
pattern is found within these species or in the other species that are used in the present study. The trajectory may instead be concave up or
down. Thus it must be stated that the work and power estimates made in
this study may contain an unknown amount of error or bias. As stated in
RESULTS, we have assumed that the
similarity in ventricular geometry and pressure waveforms among the
species included in this study would result in similar ejection
trajectories because of the law of Laplace. Analysis in terms of this
principle suggested that changes in pressure within a physiologically
realistic range (for normal individuals) during ejection would produce
only small departures from linearity of this trajectory. These changes
would result in <20% increase in the ejection work estimate.
The present
stress-ln(h0/h)
data could not be used directly to derive a constitutive or
phenomenological (predictive) equation. The advantages gained in ease
and precision of measurement would seem to make this limitation an
acceptable trade-off.
It can be argued that the stress values used here are hypothetical
because they are not measured directly (2). Pressure waves within the
lumen and anisotropy within the wall certainly cause variations in
stress among different depths and regions of the myocardium that are
beyond the analysis provided by the present approach.
The stress values are not hypothetical, however, when the mean fiber
stress over the full thickness of the myocardium in the geometric model
employed here is discussed. The overall balance of mean pressure and
mean stress in ellipsoidal shapes has been demonstrated to be almost
completely independent of the ratios of diameters (2, 26). This fact
and the fundamental law of balance of stresses in a plane (38) require
that measured pressures and wall-luminal geometries uniquely and
accurately predict values of mean fiber stress.
In summary, we have developed and tested a formalism for LV chamber and
myocardial mechanics that includes referencing of wall-thickness
measurements to calculated thickness at zero-luminal volume. This
approach allows increased direct comparison between mechanical and
energetic measurements by 1)
reducing scatter, 2) eliminating
size dependence, and 3) employing
variables that are expressed in basic physical units. Underlying
relationships are less obscured by random variations in measurements.
The fact that the approach involves only noninvasive measurements makes it practical for use in longitudinal studies and clinical applications.
| |
APPENDIX |
A cubic equation was used to calculate wall-thickness values from the
measured values of 1) end-diastolic
values of wall thickness and luminal diameter and
2) end-systolic luminal diameter. It was based on the geometric model used by Zile et al. (40) and the
approach used by Hugenholtz et al. (17). LV geometry was assumed to be
a half-ellipsoidal shell. The proportionality of the epicardial long
and short axes of this solid of revolution was assumed to be 1.2, and
the wall thickness at the apex was assumed to be 0.4 times that at the
base. In the following, L is the
endocardial long axis, D is the
endocardial short-axis, h is wall
thickness at the base, VT is total
ventricular volume (luminal + wall), and
Vc is cavity (luminal) volume. The
constant myocardial volume is
Vmyo.
The basic equations describing this model are
After
Vmyo is evaluated by using
end-diastolic dimensions, the four equations are reduced to one
|
(A2)
|
This
cubic equation is solved analytically by using classic methods.
Rearranging produces
|
(A3)
|
The
following intermediate quantities are evaluated
|
(A4)
|
The solution for h is
then
|
(A5)
|
To
find h0,
Eq. A3 is solved for the case when
D = 0
|
(A6)
|
For purposes of simplicity, a spherical ventricular geometry will be
used to illustrate the relationship between
h0/h
and the ratio of midwall dimensions. For a hemiellipsoidal geometry, such as the model of Zile et al. (40) used in this study, the result is
qualitatively identical with only coefficients changed.
An end-diastolic midwall encloses some proportion, , of the total
myocardial mass, Vmyo. The
diameter enclosing this fraction of mass at an arbitrary point in
contraction is
Dm, whereas the diameter of the lumen is D. At
zero-luminal volume, this fraction of mass would form a sphere with
diameter Dm0.
The total volume of the lumen plus myocardium can be expressed in two
ways
|
(A7)
|
This
equation can be reduced to a quadratic in
D
|
(A8)
|
Using
the quadratic formula gives an expression for
D in terms of
h and
Vmyo
|
(A9)
|
The
volume enclosed in diameter
Dm is the sum of
the luminal and a fixed proportion, , of the myocardial volume
|
(A10)
|
which
gives an expression for
Dm
|
(A11)
|
Substitution
for D using Eq. A9 gives
|
(A12)
|
or
equivalently
|
(A13)
|
At zero-luminal volume, the midwall diameter still encloses the same
proportion, , of myocardium, but no lumen, whereas the wall
thickness encloses the full myocardial volume
|
(A14)
|
These
expressions are combined to give an expression for
Dm0 in terms of
h0
|
(A15)
|
The
ratio of Dm0 to
Dm gives the
following nonlinear expression
|
(A16)
|
The
quotient on the right equals unity when
h equals
h0 and decreases
as the chamber expands. As a result of this behavior, the relationship
between mean fiber stress and
ln(h0/h)
would differ in shape from the relationship between mean fiber stress and
ln(Dm0/Dm).
| |
FOOTNOTES |
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. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: S. Denslow,
The Children's Heart Center of South Carolina, 165 Ashley Ave., PO Box
250915, Charleston, SC 29425 (E-mail: denslows{at}musc.edu).
Received 8 September 1998; accepted in final form 18 March 1999.
| |
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J APPL PHYSIOL 87(1):211-221
8570-7587/99 $5.00
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