J Appl Physiol 101: 794 –798, 2006;
doi:10.1152/japplphysiol.00865.2004.
Length and curvature of the dog diaphragm
Aladin M. Boriek,1 Ben Black,1 Rolf Hubmayr,2 and Theodore A. Wilson3
1
Baylor College of Medicine, Houston, Texas; and 2Mayo Clinic Foundation and 3Department
of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota
Submitted 10 August 2004; accepted in final form 11 May 2006
respiration; chest wall; mechanics; transdiaphragmatic pressure
THE MUSCLE BUNDLES OF THE diaphragm form a curved surface,
and transdiaphragmatic pressure (Pdi) is proportional to both
the tension in the muscle and the curvature of the muscle. As
lung volume increases, the diaphragm descends, the muscles of
the diaphragm shorten, and the pressure-generating capability
of the muscle decreases. At a given level of activation, muscle
tension decreases as the muscle shortens, and this dependence
of muscle tension on muscle length (L) is a major determinant
of the dependence of Pdi on lung volume. It has also been
argued that decreased curvature could play a role in diaphragm
failure, particularly in humans with emphysema and hyperexpanded lungs.
Measurements of Pdi and L in dogs have generated conflicting inferences about the role of curvature. In the experiments
of Road et al. (8), the rib cage was open, the abdomen was
encased in a cast with an adjustable volume, and diaphragm
length was controlled by adjusting abdominal volume. The
shape of the curve of Pdi vs. L for maximum muscle activation
matched the shape of the tension-length curve for the muscle,
and Road et al. concluded that curvature remained constant and
that the decrease in Pdi was entirely the result of decreasing
Address for reprint requests and other correspondence: A. M. Boriek, Baylor
College of Medicine, One Baylor Plaza, Dept. of Medicine, Pulmonary
Section, Suite 520B, Houston, TX 77030 (e-mail: [email protected]).
794
muscle tension. Hubmayr et al. (6) measured Pdi and L during
phrenic nerve stimulation in intact animals. They found that
Pdi decreased with shortening more rapidly than the tensionlength curve, and they concluded that decreased curvature
contributed to the fall in Pdi at large-muscle shortening. In both
of these experiments, the effect of muscle curvature on diaphragm function was inferred from the comparisons between
the curves of Pdi vs. length and tension vs. length; curvature
was not measured directly.
In our laboratory’s previous work, we measured the curvature of muscle bundles in the midcostal region of the canine
diaphragm during spontaneous breathing maneuvers (2). For
the range of muscle shortening that occurred during those
maneuvers (up to 30%), curvature was constant. To explain
this result, we presented a model in which the midcostal
diaphragm was represented as a circular arc with ends that are
fixed at the chest wall (CW). As the center of the arc descends,
the length of the arc decreases, and the radius of curvature (r)
passes through a minimum at the point where the center lies on
the line between the fixed ends. In the region of this minimum,
curvature is nearly independent of arc length. However, based
on extrapolation of this modeling, we hypothesized that curvature would decrease sharply at shortening greater than the
shortening that occurred during the spontaneous breathing
maneuvers.
Here we report the results of studies that were undertaken to
test this hypothesis. In these studies, L and curvature were
measured using the same techniques that were used in the
previous work (1, 2). Radiopaque markers were attached along
three muscle bundles in the midcostal region of the diaphragms
of six dogs. The three-dimensional positions of the markers
were calculated from the positions of the markers in biplanar
video fluoroscopic images, and L and curvature were obtained
from these data. Images were obtained at resting end expiration
[functional residual capacity (FRC)]; during inspiratory efforts
(IE) against a closed airway at FRC, FRC ⫹ 1/2 inspiratory
capacity (IC), and total lung capacity (TLC); and during
maximum phrenic nerve stimulation at the same three lung
volumes. For the spontaneous IE at different volumes, muscle
shortening, as a fraction of length at FRC, ranged from 15 to
40%. For phrenic nerve stimulation, shortening ranged from 30
to nearly 50%. We found that, for muscle shortening up to
52%, curvature is nearly constant, but for shortening greater
than 52%, curvature decreased abruptly.
METHODS
Six bred-for-research beagle dogs with body masses of ⬃8 kg were
studied, using the same methods that we have used previously (1, 2).
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8750-7587/06 $8.00 Copyright © 2006 the American Physiological Society
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Boriek, Aladin M., Ben Black, Rolf Hubmayr, and Theodore A.
Wilson. Length and curvature of the dog diaphragm. J Appl Physiol
101: 794 –798, 2006; doi:10.1152/japplphysiol.00865.2004.—Transdiaphragmatic pressure is a result of both tension in the muscles of the
diaphragm and curvature of the muscles. As lung volume increases,
the pressure-generating capability of the diaphragm decreases.
Whether decrease in curvature contributes to the loss in transdiaphragmatic pressure and, if so, under what conditions it contributes are
unknown. Here we report data on muscle length and curvature in the
supine dog. Radiopaque markers were attached along muscle bundles
in the midcostal region of the diaphragm in six beagle dogs of ⬃8 kg,
and marker locations were obtained from biplanar images at functional residual capacity (FRC), during spontaneous inspiratory efforts
against a closed airway at lung volumes from FRC to total lung
capacity, and during bilateral maximal phrenic nerve stimulation at
the same lung volumes. Muscle length and curvature were obtained
from these data. During spontaneous inspiratory efforts, muscle shortened by 15– 40% of length at FRC, but curvature remained unchanged. During phrenic nerve stimulation, muscle shortened by 30 to
nearly 50%, and, for shortening exceeding 52%, curvature appeared to
decrease sharply. We conclude that diaphragm curvature is nearly
constant during spontaneous breathing maneuvers in normal animals.
However, we speculate that it is possible, if lung compliance were
increased and the chest wall and the diameter of the diaphragm ring of
insertion were enlarged, as in the case of chronic obstructive pulmonary disease, that decrease in diaphragm curvature could contribute to
loss of diaphragm function.
LENGTH AND CURVATURE OF DIAPHRAGM
other investigators (3, 6, 7, 8). Briefly, the spinal roots of the phrenic
nerves (C5, C6) were identified and isolated on both sides of the neck.
Insulated hook electrodes were placed under the nerve roots, and the
preparation was covered with mineral oil. Supramaximal synchronous
tetanic stimulations at frequencies between 1 and 50 Hz were applied
to the nerve roots using a Grass S 88 nerve stimulator. The tetanic
stimulation parameters used are the following: amplitude (6 V), pulse
duration (1 ms), delay between pulses (0.01 ms), pulse frequency
(1–50 Hz), and pulse train duration (500 ms). We recorded biplane images in the supine position before and after bilateral stimulation at lung volumes spanning the vital capacity: FRC, FRC ⫹ 1/2
IC, and TLC.
Data analysis. The coordinates of the markers in the two biplane
images for each maneuver were determined, and the three-dimensional coordinates of the markers were computed. Examples of anterior-posterior views of the marker locations are shown in Fig. 1, left.
It was apparent from plots like that shown in Fig. 1, bottom left, that
the markers on the CW did not lie on the curve drawn through the
other four markers on each muscle bundle. In addition, the distance
between the markers on the CW and the adjacent markers along a
muscle bundle decreased less than the distances between other markers. These discrepancies suggest that the markers on the CW lay
caudal to the effective line of insertion of the diaphragm. Defining the
line of insertion is not straightforward. The diaphragm is anchored to
the rib cage along a line that lies cranial from the line of attachment
to the abdominal muscles. We placed the markers along the line
connecting the diaphragm to the abdominal muscles on the abdominal
side of the diaphragm, and it appeared that this line lay caudal to the
Fig. 1. Examples of the data on marker locations for one
dog at functional residual capacity (FRC) (top) and during
bilateral phrenic nerve stimulation with the airway occluded
at FRC (bottom). In each panel, the anterior-posterior view
of the data in laboratory coordinates (x, y, z) is shown on the
left. The z-axis lies in the cranial direction, and the y-axis in
the lateral direction. The muscle bundles extend from the
insertion on the central tendon (CT) to the insertion on the
chest wall (CW). The view in the coordinate system (␰, ␩,
␨) of the plane fit to the data is shown on the right. ␩ is
perpendicular to the ␰, ␨, plane. The ␨-axis lies in the
direction perpendicular to the plane fit to the markers, and
the ␰-axis lies in the direction of maximum curvature. The
scale of the ␨-axis is expanded relative to that of the
abscissa. Circular arcs fit to the data are shown by dashed
lines.
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Dogs were maintained according to the National Institutes of Health
“Guide for the Care and Use of Laboratory Animals,” and all procedures were approved in advance by the Institutional Review Board of
Baylor College of Medicine and Mayo Clinic. Radiopaque markers
were attached to the diaphragm by the following procedure. The
abdomen was opened by midline laparotomy, and 2-mm siliconcoated beads were stitched to the peritoneal surface of muscle bundles
in the midcostal region of the left diaphragm. Five markers were
placed along each of three nearby muscle bundles: one at the origin of
each muscle bundle on the central tendon (CT), one at its insertion on
the CW, and three at equal intervals along the muscle bundle. The
animals were allowed to recover for at least 3 wk.
The animals were anesthetized with pentobarbital sodium (30
ml/kg), intubated with a cuffed endotracheal tube, and placed in the
test field of a biplanar fluoroscopic system. This high spatial (⫾0.5
mm) and temporal (30 Hz) resolution system was used to record
displacement of the radiopaque metallic markers. The animal was
mechanically ventilated to apnea, and IC was measured by inflating
the animal to TLC, defined as the volume at an airway pressure of 30
cmH2O. The animal was allowed to resume spontaneous breathing,
and biplane images were obtained at end expiration. The airway was
then occluded at FRC, or the lungs were inflated and the airway was
occluded at either FRC ⫹ 1/2 IC or TLC. The change in airway
opening pressure (⌬Pao) was measured during IE against the occluded
airway, and, when ⌬Pao reached a plateau, usually during the fifth or
sixth IE, biplane images were recorded at the peak of the IE.
In vivo phrenic nerve stimulation. The roots of the phrenic nerve
were stimulated using the same technique as that used by a number of
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LENGTH AND CURVATURE OF DIAPHRAGM
L ⫽ a␾/sin ␾ ⫺ c
(1)
r ⫽ a/sin ␾
(2)
For given values of a and c, these equations describe r as an implicit
function of L.
Statistical analysis. Based on the mathematical model described in
Eqs. 1 and 2, we simulated a data set and generated two statistical
models. Figure 3 shows lines plotted for the mathematical model as
well as the two statistical models. The mathematical model was fit to
the data by defining the average values for the two ends of the range
of L at FRC (LFRC) and solving for the constants a and c in Eqs. 1 and
2. The first statistical model was found by generating a complex
function that best fit the relationship between L/LFRC and r as defined
by Eqs. 1 and 2, with an adjusted error value of ⬍1%. The second
statistical model shows the same form of an equation fit to the raw
data of L/LFRC and r. Figure 3 shows the three models plotted as lines
over the raw data.
RESULTS
Individual values of LFRC, length as a fraction of LFRC, and
r are listed in Table 1. The r of the muscle bundles vs. L as a
fraction of LFRC for a ⫽ 4.4 cm, c ⫽ 1.6 cm, and LFRC ⫽ 5.3
cm is plotted in Figure 3. In the mathematical model and in the
statistical models, muscle shortening (1 ⫺ L/LFRC) is always
J Appl Physiol • VOL
Fig. 2. Model of the kinematics of the midcostal region of the diaphragm. In
this region, the diaphragm is modeled as an arc of a circular cylinder that lies
between fixed points on the CW that lie at a distance a from the midplane of
the circular cylinder. The arc is made up of muscle of variable length L that lies
between the origin at CW and the insertion at the CT and a fixed length c of
CT that lies between CT and the midplane. As the center of the arc (O)
descends, L decreases, and the radius of the arc r increases. According to the
model, curvature is sensitive to the size of the ring of insertion that is described
by the parameter a. If a were increased by 10% and c and LFRC were
unchanged, r at FRC would increase by 10%, but the point at which r increases
sharply would shift from L/LFRC ⬃ 0.52 to L/LFRC ⬃ 0.62, where LFRC is
length at FRC.
above 48%. More specifically, r is approximately constant for
L/L0 ⬎ 0.52 and increases sharply for L/L0 ⬍ 0.52. On the
other hand, the data-based model takes into account the inherent variability in the data and has a more conservative estimate
of the location of the break point, the point at which the radius
rises sharply. It predicts that r is approximately constant for
L/L0 ⬎ 0.517 and increases sharply for L/L0 ⬍ 0.517. The
measurements came from six different dogs, and the fit of the
curves representing the individual dogs are not shown; however, none of the data points fell to the left of the minimum
value of L/LFRC, and the general shape of the relationship
did not contradict the fit shapes for individual dogs. We
conclude that the data-based model is conservative enough for
the purpose of estimating a statistically significant rise in r for
L/L0 ⬍ 0.52.
DISCUSSION
Our objective in this study was to test the hypothesis that
diaphragm curvature would decrease at muscle shortening
greater than the shortening that occurred during the spontaneous breathing maneuvers that we studied in our previous work.
We obtained data on diaphragm curvature as a function of
muscle length for more extreme muscle shortening, and we
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effective line of insertion on the CW. We, therefore, ignored the data
for the markers on the CW in the subsequent analysis. The lengths of
each muscle bundle were determined by adding the three distances
between the four markers on each muscle bundle. Lengths during IE
and phrenic nerve stimulation were expressed as fractions of the
length at FRC, and the values for the three bundles were averaged.
A plane was fit to the coordinates of the 12 markers, and the marker
coordinates were transformed to a coordinate system based on that
plane with the ␰⬘ and ␩⬘ axes of the coordinate system in the plane and
the ␨ direction normal to the plane. The plane was fit to all markers,
including those on the CW, by minimizing the sums of the squares of
the distances between the marker locations and the plane. The marker
coordinates in a new coordinate system with the ␩ and ␨ axes in the
plane and the ␰-axis normal to the plane were determined. A quadratic, ␰ ⫽ a␰2 ⫹ b␰⬘␨⬘ ⫹ c␨ ⫹ d␰2 ⫹ e␨2 ⫹ F, was fit to the data by
a multiple least squares regression technique. Principal coordinates ␰,
␩, and ␨⬘ were determined, for which the quadratic has the form, ␰ ⫽
1/2C1 ␩ ⫹ 1/2C2 ␨ ⫹ F⬘. The coefficients C1 and C2 were the
principal curvatures of the quadratic fit to the data. The ␰ was chosen
as the coordinate with the largest curvature. The data were then
transformed to a third coordinate system (␰, ␩, ␨), in which the ␰-axis
lay along the direction of maximum curvature. Examples of the
marker locations in this coordinate system are shown in Fig. 1, right.
For most conditions, the line of the muscle bundles lay nearly parallel
to the direction of maximum curvature, the ␰ direction, and the
curvature in the transverse direction, the ␩ direction, was small. In
those cases, the r is the inverse of the curvature in the ␰ direction. In
a few cases, the line of the muscle bundle lay at an angle to the ␰-axis.
In those cases, the curvature along the muscle bundle and the corresponding r were used.
Modeling. A simplified version of the model for diaphragm kinematics that our laboratory presented in an earlier paper (2) is shown in
Fig. 2. In this model, the shape of the midcostal region of the
diaphragm is pictured as being an arc of a circular cylinder that
extends between two fixed points on the CW that lie at distances a
from the midplane of the cylinder. The center of the circular arc lies
at point O on the midplane, and the arc of the diaphragm between the
CW and the midplane subtends an angle ␾. This arc is composed of
muscle with variable length that lies between CW and CT, and CT
with fixed length c that lies between CT and the midplane. The L and
the r are related to a, c, and ␾ by the following equations.
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LENGTH AND CURVATURE OF DIAPHRAGM
Fig. 3. Radius of curvature of the muscle bundles vs. muscle L, as a fraction
of LFRC. Data for 32 conditions for a total of 6 dogs are shown by different
symbols for each of the 6 dogs. The figure shows the radius of curvature r
plotted as a function of L/LFRC for the 6 dogs studied. The lines plotted on top
of the raw data show 3 models of the relationship between r and L/LFRC: the
mathematical model described in Eqs. 1 and 2, a statistical model in a specific
form fit to that mathematical model, and a statistical model of the same specific
form fit to the raw data.
J Appl Physiol • VOL
Table 1. Muscle lengths and radii of curvature
IE
FRC
Dog No.
FRC
FRC ⫹ 1/2 IC
L, cm
MPNS
TLC
FRC ⫹ 1/2 IC
TLC
0.55
0.59
0.53
0.60
0.54
0.56
⫾0.03
0.55
0.54
0.52
0.55
⫾0.03
5.2
4.9
4.1
20.7
6.3
5.8
9.5
⫾7.5
27.6
5.2
6.3
11.0
⫾11.1
17.3
13.5
22.9
14.5
⫾7.9
FRC
L/LFRC
1
4.7
0.87
2
4.5
0.89
3
4.9
0.84
4
4.0
0.86
5
4.2
0.85
6
4.1
0.82
Average 4.4
0.86
SD
⫾0.358 ⫾0.02
0.62
0.77
0.68
0.70
0.71
0.61
0.68
⫾0.06
0.65
0.63
0.72
0.61
0.60
0.55
0.80
0.70
0.66
0.54
0.67
0.61
⫾0.08 ⫾0.08
r, cm
1
6.3
2
4.4
3
4.2
4
4.0
5
3.7
6
3.1
Average 4.3
SD
⫾1.1
3.5
2.6
2.8
4.8
4.6
4.5
4.1
⫾0.9
3.5
5.2
8.4
7.0
4.0
4.2
4.7
⫾1.3
6.4
2.6
3.8
4.2
3.8
3.7
4.0
⫾1.3
FRC, functional residual capacity; IE, end of inspiratory effort; MPNS,
maximal phrenic nerve stimulation; TLC, total lung capacity; IC, inspiratory
capacity; L, diaphragm muscle length; LFRC, L at FRC; r, radius of diaphragm
curvature; SD, standard deviation.
dependence of curvature on other parameters. In particular,
curvature is sensitive to the diameter of the ring of insertion
that is described by the parameter a. If a were to increase by
10% and the other parameters, c and LFRC, were unchanged, r
at FRC would increase by 10%, but, more importantly, the
point at which r increases sharply would shift from L/LFRC ⬃
0.52 to L/LFRC ⬃ 0.62.
Pdi is proportional to both muscle tension and curvature.
Because curvature is nearly constant for L/LFRC ⬎ 0.52, the
curve of Pdi vs. L/LFRC would have the same shape as the curve
of muscle force vs. L/LFRC for L/LFRC ⬎ 0.52. Because curvature decreases sharply for L/LFRC ⬍ 0.52, the curve of Pdi
vs. L/LFRC would be steeper than the force-length curve for
L/LFRC ⬍ 0.52. This behavior is consistent with the data for Pdi
vs. L/LFRC reported by Hubmayr et al. (6).
Diaphragm muscle shortening of ⬎40% would not occur in
normal dogs during spontaneous breathing. In particular, our
data demonstrated that muscle shortening was less during
spontaneous breathing than that during phrenic nerve stimulation (P ⬍ 0.05). The high degree of muscle shortening that
occurred during maximal phrenic nerve stimulation is nonphysiological for two reasons. First, the diaphragm is never
maximally activated during spontaneous respiratory efforts (5).
Second, during phrenic nerve stimulation in our preparation,
the muscles of the rib cage were silent. If these muscles were
active, as they are during spontaneous breathing, the load on
the diaphragm would be greater and diaphragm descent and
muscle shortening would be smaller than they were in our
experiments. However, in chronic obstructive respiratory disease, lung compliance is increased so that the load on the
diaphragm is decreased (9, 11), and the rib cage is enlarged so
that the diameter of the ring of insertion is increased (4, 10,
12). In that situation, a decrease in diaphragm curvature could
contribute to a loss in diaphragm function.
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modified our model of diaphragm kinematics to explain the
relation between L and curvature for the range of muscle
shortening that was tested.
L and curvature were measured by the radiopaque marker
technique, and this technique is well suited to the objectives of
measuring L and curvature. The location of the markers is
measured to within an accuracy of ⫾0.5 mm, and this precision
is needed to obtain accurate values of the r. As can be seen in
Fig. 1, at FRC, the positions of the markers deviate from the
best fit plane by distances of the order of 2–3 mm. The
distances of the markers from the plane constitute the signal
from which curvature is obtained. Thus, at FRC, the signal-tonoise ratio is small. The uncertainty in the curvature of the
quadratic fit to the data at FRC is ⫾10%.
Given the relationship of curvature and its radius: [␬] ⫽ 1/r,
if the range of [␬] is (0.9 [␬]0, 1.1[␬]0), the uncertainty of r
ranges between 0.91 r0 and 1.11 r0, where r0 ⫽ 1/[␬]0. During
phrenic nerve stimulation, the absolute magnitude of the uncertainty in the quadratic fit is about the same, but, because the
curvature is smaller, the normalized relative uncertainty of r
(divided by r0) is the same, although the magnitude of the
uncertainty is bigger.
The primary results of this study, as shown in Fig. 3, are the
following. For muscle shortening of up to ⬃52%, the r of the
muscle bundles appears to remain nearly constant, and for
muscle shortening ⬎52%, r appears to increase sharply. We
explained the relation between r and L/LFRC in a simple model
for diaphragm kinematics. In such model, the diaphragm is
pictured as lying on the arc of a circular cylinder that extends
between fixed end points. As the center of the arc descends, the
muscle shortens, curvature remains nearly constant until
L/LFRC reaches ⬃0.52, and then curvature decreases sharply
for greater shortening. The model can be used to predict the
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LENGTH AND CURVATURE OF DIAPHRAGM
ACKNOWLEDGMENTS
We thank Dr. Saleh Al-Muhammed and Dr. Deshen Zhu of Baylor College
of Medicine for conducting the surgery and for technical assistance. We also
thank Bruce Walters of the Mayo Clinic for very valuable technical assistance.
GRANTS
This work was supported in part by the National Heart, Lung, and Blood
Institute Grants HL-072839 and HL-63134.
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