Vol. 280, Issue 2, R588-R597, February 2001
Modeling the kinematics of the canine midcostal diaphragm
Maneesh R.
Amancharla,
Joseph R.
Rodarte, and
Aladin M.
Boriek
Baylor College of Medicine, Houston, Texas 77030
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ABSTRACT |
The
hypotheses that the chest wall insertion (CW) is displaced laterally
during inspiration and that this displacement is essential in
maintaining muscle curvature of the costal diaphragmatic muscle fibers
were tested. With the use of data from three dogs, caudal, lateral, and
ventral displacements of CW during both quiet, spontaneous inspiration
and during inspiratory efforts against an occluded airway were observed
and recorded. We have developed a kinematic model of the diaphragm that
incorporates these displacements. This model describes the motions of
the muscle fibers and central tendon; the displacements of the
midplane, muscle-tendon junction (MTJ), CW, and center of the muscle
fiber-central tendon arcs are modeled as functions of muscle fiber
length. In the model, the center of the fiber arcs and MTJ both
move caudally parallel to the midplane during inspiration, whereas CW
moves both caudally and laterally. The observed lateral displacement of
CW and the observed caudal displacement of MTJ, as functions of muscle
fiber length, both approximate well the theoretical displacements that would be necessary to maintain curvature of the fiber arcs. In confirming our hypotheses, we have found that lateral displacement of
CW is a mechanism by which changes in the shape of the costal diaphragm, as described by its curvature, are limited.
respiratory mechanics; chest wall; muscle
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INTRODUCTION |
WE HAVE
PREVIOUSLY shown that the muscle fibers (MFs) of the active
canine midcostal diaphragm form arcs of circles whose curvature varies
little in the physiological range of lung volumes, i.e., from
functional residual capacity (FRC) to end of inspiratory effort at
total lung capacity (TLC) (3, 4). Given that the MFs contract by ~35% of their length in the physiological range, some mechanisms are at work by which the diaphragm maintains shape. These mechanisms, however, are poorly understood. Whereas a previous study showed that diaphragm shape is maintained as a pressurized membrane due to muscle anisotropy and central tendon (CT)
inextensibility (2), another mechanism is sought because
anisotropy does not fully explain the effects of muscle shortening and
activation on diaphragm shape. This study examines displacement of the
chest wall insertion (CW) as a mechanism that limits changes in
diaphragm shape during inspiration
It is useful first to consider the mechanical properties of the
diaphragm. The MFs and CT, which together comprise the diaphragm, form
a thin, dome-shaped sheet: a membrane (8). As a
pressurized membrane, the diaphragm cannot carry bending moments or
compressive stresses in its plane; continuity of slope on the diaphragm
surface is therefore expected. The only significant stresses that the diaphragmatic membrane could carry are tensile and shear stresses in
the plane of its surface. Maintenance of diaphragm shape also supports
the likely scenario of uniformity in stress along the length of an MF.
The CT is essentially isotropic and inextensible within the range of
physiological stresses imposed on it (2).
A number of assumptions regarding the shape and motion of the diaphragm
must also be made when modeling its motion. First, it should be
emphasized that only the midcostal region of a hemi-diaphragm is being
modeled and that any assumptions made do not necessarily extend to the
dorsal costal or crural regions of the diaphragm. The diaphragm
exhibits uniform motion during inspiration. The MF bundles lie along
curved lines and attach to CW and CT. The MFs contract during
inspiration, generating stresses parallel to their length. Each MF arc
moves in its own best-fit plane of maximum curvature during inspiration
(1), and the centers of the circles associated with these
arcs move along a specific trajectory (the nature of this trajectory
will be discussed). Because MFs lie along lines of maximum principal
curvature of the surface of the diaphragm, the terms "in the
direction of an MF" and "in an MF's plane of maximum curvature"
are similar (the former is a line in the plane of the latter). Planes
of maximum curvature for different MFs need not be parallel to each
other, although they do appear to be parallel for adjacent muscle
bundles in the midcostal diaphragm (3, 4).
In the midcostal region, the curvature of the diaphragm transverse to
the MFs is not significantly different from zero. Boriek et al.
(4) have illustrated this property by orienting adjacent midcostal MFs in their own planes of maximum curvature; as a result, the fibers appear to be superimposed on each other at any lung volume
during inspiration. Thus adjacent midcostal MFs together form a section
of a cylinder whose long axis is parallel to the CW insertion (1,
3).
Although more physiologically accurate than previous models of
diaphragm motion (5-7), the model of diaphragm
kinematics proposed by Boriek et al. (3) did not consider
certain properties of the diaphragm, most notably, displacement of the
CW. In this paper, we propose a model of costal diaphragm kinematics
expanded from that of Boriek et al. (3). This model
includes lateral displacement of CW as an essential mechanism by which
changes in the shape of the active midcostal diaphragm, as described by its muscle fiber curvature, are limited.
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METHODS |
Three bred-for-research beagle dogs, weighing between 9.9 and
10.5 kg, were surgically prepared using the same methods that were used
in an earlier study (1). The abdomen was opened by midline
laparotomy, and 2-mm silicon-coated lead spheres were stitched to the
peritoneal surface of muscle bundles in the midcostal region of the
left hemidiaphragm and to the peritoneal surface of CT along the
midline (Fig. 1). Four markers were
placed along each of three nearby muscle bundles: one at the origin of
each muscle bundle on CT ["muscle-tendon junction (MTJ) marker"],
one at its insertion on CW ("CW marker"), and two at equal
intervals along the muscle bundle. Seven markers were stitched to CT
along the midline from the sternum to the spine ("midplane
markers"). The animals were allowed to recover from the surgery for
at least 3 wk.
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Fig. 1.
Locations of metallic markers on abdominal surface of
left midcostal diaphragm and midplane. Four markers were sutured along
each of 3 muscle bundles from the origins of the bundles on the central
tendon (CT) to their insertions on the chest wall (CW). Seven
additional markers were sutured to the midplane of the diaphragm: 4 on
the costal diaphragm, 3 on the crural. MTJ, muscle-tendon junction.
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The animals were anesthetized with pentobarbital sodium (30 mg/kg),
intubated with a cuffed endotracheal tube, and placed in the supine or
prone posture in a radiolucent body plethysmograph that was situated in
the test field of a computer-based biplanar videoroentgenographic
recording system. This high spatial (±0.5 mm) and temporal (30 Hz)
resolution system was used to record displacements of the radiopaque
metallic markers. Balloon-tipped catheters were inserted in the stomach
and esophagus. The positions of the catheters were checked by
fluoroscopy and by demonstrating that abdominal pressure increased and
esophageal pressure decreased during a spontaneous breath and that
esophageal and airway pressures decreased equally during an occluded
inspiratory effort at FRC. Biplanar images were recorded continuously
during five spontaneous breaths. The lungs were then inflated and
occluded successively at various lung volumes. The airway was held
occluded until the animal made inspiratory efforts against the occluded
airway. Biplanar images were recorded when the change of airway
pressure reached a plateau, usually during the fifth or sixth
inspiratory effort at each lung volume. The animal was then rotated to
the opposite posture, and the procedure was repeated.
We therefore investigated both 1) quiet, spontaneous,
open-airway breathing (i.e., normal lung volume expansion occurs) and 2) spontaneous inspiratory efforts against an occluded
airway (i.e., lung volume held constant). "Inspiration" thus refers
both to quiet, spontaneous breathing and to inspiratory efforts against an occluded airway, both of which are included in our data. During the
quiet, spontaneous breathing maneuvers, images were captured at two
lung volumes: end of expiration (EE) and end of inspiration (EI).
Similarly, during the occluded airway maneuvers, images were captured
at the end of inspiratory effort at three occluded lung volumes: FRC,
FRC + one-half inspiratory capacity (IC), and TLC.
The coordinates of the lead spherical markers in the two biplanar
images were determined, and the three-dimensional coordinates of the
markers were computed at all lung volumes. The length of each muscle
bundle in each state was determined by adding the distances between
markers on each bundle, and the average length of each of the three
bundles across five consecutive efforts was computed. This incremental
linear fit to the curved muscle bundle is valid because muscle bundles
form smooth arcs, and the four markers on a muscle bundle were placed
only 1-2 cm apart. The position of the markers relative to the
muscle is ensured because the markers were stitched onto the muscle.
Finally, at the end of the experiment, the diaphragm was excised and
the configuration of the markers was confirmed visually.
A plane was fit through the locations of the 12 markers in the
midcostal diaphragm at occluded TLC. This plane was used as the basis
for a local 
-
-
coordinate system in which the data for all
lung volumes for a given dog and posture were viewed (Fig. 2). The plane of maximum curvature of
these 12 markers was computed and defined as the - plane. The
midplane of the dog was determined by fitting a plane through the
locations at TLC of the seven markers stitched along the midline on CT.
The - axes were rotated in their own plane to align the -axis
with the midplane of the dog. The -axis is perpendicular to the
- plane. The data from all lung volumes for a given dog and
posture were then transformed to the -- coordinate system and
viewed in the - plane. In the midcostal region of the diaphragm,
positive values of and are only slightly different from the
conventional anatomic coordinates "ventral" and "lateral,"
respectively, and will be referred to as such. Positive will be
referred to as cranial, and negative will be referred to as caudal.
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Fig. 2.
Orientation of the - plane (i.e., plane of maximum curvature
of muscle bundles of the left midcostal diaphragm) is shown relative to
the anatomic Cartesian coordinate system (x, y,
z). The CW insertion extends from spine (SP) to sternum
(ST). The muscle bundle drawn extends from its origin at the MTJ to its
insertion on the CW.
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Displacements of CW, MTJ, and midplane markers both in and out of the
plane of maximum curvature (- plane) were examined. Of the seven
midplane markers, we excluded the three most dorsal and the single most
ventral markers and considered only the three remaining markers because
they were closest to a continuation of the midcostal MFs. These three
markers were then used to calculate the displacement of the midplane.
The data analyzed in this study were originally obtained from
experiments in a previously published work (3). However, the majority of the data used for this study was not considered in that
earlier work. The earlier work analyzed displacements of the MF and MTJ
but dismissed displacements of CW as negligible; this study focuses on
the displacements of CW. Also, data from the midplane markers on the CT
are considered for the first time in this study.
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RESULTS |
Displacements and equations.
We have observed the motion of the CW, MTJ, and midplane markers during
inspiration. Visual inspection of the data reveals that the insertion
on CW is not fixed. Further mathematical analysis of the markers
confirms displacements of CW, MTJ, and the midplane during inspiration.
Displacements for each posture along each coordinate axis are recorded
from the midcostal CW markers (Fig. 3),
from the midcostal MTJ markers (Fig. 4),
and from the midplane markers (Fig. 5).
The displacements in these figures represent displacements from EE to
EI during quiet, spontaneous breathing, from EE to spontaneous effort
at an occluded lung volume of FRC, from EE to spontaneous effort at
occluded FRC + one-half IC, and from EE to spontaneous effort at
occluded TLC.
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Fig. 3.
CW displacement ( CW) relative to end of expiration
(EE) at various lung volumes. The error bars represent SE.
A: positive values of CW represent ventral
displacement of the CW. B: positive CW
represents lateral displacement. C: positive
CW represents cranial displacement. FRC, functional
residual capacity; IC, inspiratory capacity; TLC, total lung capacity;
EI, end of inspiration.
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Fig. 4.
MTJ displacement (MTJ) relative to EE at various
lung volumes. The error bars represent SE. A: positive
values of MTJ represent ventral displacement of MTJ.
B: positive MTJ represents lateral
displacement. C: positive MTJ represents
cranial displacement.
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Fig. 5.
Midplane displacement, relative to EE, at various lung
volumes. Of the 7 midplane markers, the 3 on the CT and costal
diaphragm were chosen to calculate these displacements. The error bars
represent SE. A: positive values of
midplane represents ventral displacement of the
midplane. B: positive midplane represents
lateral displacement. C: positive midplane
represents cranial displacement.
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We have best fit a line through the data points in displacement plots
of CW, MTJ, and midplane versus normalized muscle length 
( = 1 at EE), considering displacements on each axis separately. Thus
equations of CW, MTJ, and midplane displacements as functions of
normalized muscle length have been determined and are listed in Tables
1-3
(all numbers are in cm; positive values of , , and represent
ventral, lateral, and cranial motions, respectively).
Interestingly, the motion of the diaphragm from EE to EI (i.e., from a
passive to an active state) appears to be qualitatively different from
its motion between active states. Also, the MF markers at EE are fit
less well by an arc than the markers at other lung volumes, as can be
seen in Fig. 6A, which shows
the positions of the diaphragm markers in the - plane for one dog in the supine posture. An observation of the motion of the markers on
the diaphragm as it moves from an active to a passive state at the same
lung volume suggests that the motion of the diaphragm is quite
different during that brief transition from an active to a
passive state. An inspection of the equations for CW displacement (Table 1) indicates that there is a higher rate of CW displacement with
respect to unit MF shortening at low lung volumes during the transition
from a passive to an active state at the onset of an inspiratory
effort. The focus of this paper, however, is the kinematics of the
active diaphragm.
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Fig. 6.
Locations of the muscle fiber (MF) and midplane markers,
for a supine dog, viewed in the - coordinate system (positive is lateral; positive is cranial). A: the data are shown
along with best- fit curves. On the right, an arc of a circle is fit to
the MF markers at each lung volume; on the left, a straight line is fit
to the trajectory of each of the 4 midplane markers as lung volume
changes. B: from the data in A, the best-fit arcs
to the MFs at each lung volume are redrawn. The separate trajectories
of the 4 midplane markers are combined, and an average midplane
trajectory is shown. Additionally, the average CW and MTJ locations at
each lung volume and the best-fit line to the MTJ markers are shown.
The angle between the trajectories of the midplane and MTJ markers is
~7°.
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Ventral motion.
To determine whether the ventral (perpendicular to the plane of maximum
curvature of the MFs) displacement of CW represents a uniform ventral
displacement of the midcostal diaphragm, the ventral displacements of
CW, MTJ, and the midplane are compared: ratio of MTJ
to CW to midplane = 1.78:1.87:1.00 in the prone posture, and 0.82:2.07:1.00 in the supine
posture, where is change. Because the ventral displacements
of the CW, MTJ, and midplane markers are on the same order of
magnitude, it can be inferred that a uniform ventral displacement of
the entire diaphragm takes place. Uniform dorsoventral motion of the diaphragm suggests an absence of torsion of the diaphragm surface during inspiration. Teleologically, an absence of torsion increases efficiency because energy is used in displacing the diaphragm rather
than wasted in distorting its surface.
MTJ and midplane.
Visual inspection of Figs. 3 and 4 suggests that the displacement, on
each axis, of the MTJ markers approximates that of the midplane
markers. Two-sample t-tests (2 tailed, 95% confidence level) confirm that there are no significant differences between the
displacements, on any axis, of the MTJ and midplane markers (comparing
each posture and lung volume separately).
As shown in Figs. 3 and 4 and in Tables 2 and 3, the motions both of
the MTJ and midplane markers are primarily caudal and slightly ventral.
We have already examined the ratios of ventral displacements and have
found that a uniform ventral displacement of the entire midcostal
diaphragm occurs. Examining the ratios of caudal displacements of the
MTJ and midplane markers shows only small differences between the two:
MTJ-to-midplane ratio = 1.21:1 (prone) and 1.47:1 (supine). These small
differences are partly attributable to the difficulty in stitching
precisely, in vivo, the midplane markers onto the hypothetical
extension of an MF on the CT (i.e., intersection of the - plane
and the midplane). Moreover, Tables 2 and 3 show that the caudal
displacements both of the MTJ (prone: R2 = 0.982; supine: R2 = 0.967) and midplane
(prone: R2 = 0.991; supine:
R2 = 0.992) are highly regular and
predicted well by a linear fit to the data.
There is an absence of consistent lateral/medial displacement as a
function of muscle length of both the MTJ (prone:
R2 = 0.248; supine:
R2 = 0.225) and midplane (prone:
R2 = 0.063; supine:
R2 = 0.685) markers. Furthermore, a
two-sample t-test (2 tailed, 95% confidence interval)
indicates no significant differences between the lateral displacements
of both the midplane and MTJ markers. Thus the trajectory of the MTJ
markers during inspiration may be parallel to the midplane.
Figure 6 illustrates the positions and trajectories of the diaphragm
markers for one dog in the supine posture. In Fig. 6A, midplane markers at the same lung volume appear to form an arc; this is
the case because they were stitched onto the dome-shaped central tendon
and are now viewed in the plane of maximum curvature of the MFs, which
is not precisely perpendicular to the plane of maximum curvature of the
midplane markers on the CT. Thus, although different midplane markers
at the same lung volume form an arc, the trajectory of each midplane
marker is approximately linear. It is these trajectories that are drawn
in Fig. 6A and whose average is shown in Fig. 6B.
One can see that the trajectory of the MTJ during inspiration is
essentially linear and parallel to the trajectory of the midplane
markers; these two linear trajectories form a 7° angle for this dog.
Chest wall.
Figure 3 and Table 1 reveal caudal and lateral displacements of the
chest wall during inspiration. Although not as linear as the caudal
motion of the MTJ and midplane, a linear fit to the caudal displacement
of the CW versus muscle length is quite good (prone:
R2 = 0.623; supine:
R2 = 0.793). The lateral displacement of
the CW, however, is only fit well by a line in the prone posture
(R2 = 0.869); a linear fit does not predict
the data well in the supine posture (R2 = 0.218). Our model of midcostal diaphragm kinematics is thus better
supported by the physiological prone posture than by the unphysiological supine posture.
T-tests (2 tailed, 95% confidence level) reveal significant
differences between the prone and supine postures for all three components of CW displacement and for the ventral and caudal (but not
the lateral/medial) components of MTJ displacement. No significant differences have been found between the postures for the midplane displacements.
Kinematic model.
In an earlier work, Boriek et al. (4) did observe caudal
and lateral displacement of CW, but these displacements were not computed and were not included in the kinematic model of diaphragm mechanics that they later formulated (3). We have computed these CW displacements and incorporated them into a kinematic model of
the canine midcostal diaphragm.
Our kinematic model of the midcostal diaphragm is shown in Fig.
7. An MF and its CT extension are shown
in its - plane. The MF is represented by the solid arc from CW to
MTJ; the angle spanned by the MF is denoted 
. The CT is represented
by the solid arc from MTJ to the midplane; the angle spanned by the CT
is denoted 
. The radius of the MF-CT arc is R; the center
of the arc is C. As the MF shortens, both the midplane and
center of the MF-CT arc move caudally along AB, and the MTJ
moves in a parallel fashion along DE. CW moves on
FG.
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Fig. 7.
Model of diaphragm kinematics. Solid arcs represent the
MF and CT; dotted lines represent trajectories; dashed lines set off
angles. Arc AD is the CT; arc DF is the MF.
AB is the midplane. In this model, as the fiber shortens, a
point on the midplane and the center of the MF-CT arcs (C) move on
AB. The MTJ moves on DE, and the CW moves on
FG. Note, for reasons of clarity, the vector on
FG representing displacement of CW from FRC to TLC is shown
3 times larger than it should be in relation to the lengths of the MF
and CT.
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The lateral displacement of CW [CW()] is
predicted by the model and found experimentally, whereas the caudal
displacement of CW [CW()] is only found
experimentally. Caudal displacements of the midplane
[midplane()] and MTJ
[MTJ()] with respect to that of the chest wall
[CW()] are both predicted by the model and found
experimentally. The model predicts no lateral/medial motion of the MTJ
and midplane, and the experimental data confirm this. Finally, the
uniform ventral (+) displacement of the entire diaphragm suggested
by the data is not predicted by, but is consistent with, the model.
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DISCUSSION |
MTJ and midplane.
The highly linear, parallel, and approximately equal caudal and ventral
motions of the MTJ and midplane, their statistically insignificant
lateral/medial motions, and the lack of statistically significant
differences among their motions on any axis together suggest that the
MTJ should always have the same location relative to a point on the CT
that lies on the midplane, provided that both are in the same -
plane. This assertion is also made likely by the inextensibility of the
CT and the fact that points on the CT that are on the midplane at one
lung volume probably remain on the midplane at other lung volumes.
The absence of significant lateral/medial motion of the midplane
markers is also predicted by the mechanical properties of the
diaphragm: lateral/medial motion of the midplane would require either
torsion of the diaphragm surface or asymmetry in the motions of the
left and right hemidiaphragms. If the midplane were to move laterally
for MFs in one hemidiaphragm, it would move medially for MFs in the
other hemidiaphragm. But if CW were fixed and if curvature in the
direction of an MF were maintained, this phenomenon could occur only if
the angle subtended by the MF and CT from CW to midplane is <90° in
one hemidiaphragm and >90° in the other. This asymmetry, in which
the midcostal MFs are of different lengths in opposite hemidiaphragms
and in which the midplane follows different trajectories in different
fibers' planes of maximum curvature, is unlikely.
Chest wall.
A fixed CW is inconsistent with the kinematics and mechanics of the
canine midcostal diaphragm. Key assumptions regarding the kinematics
and mechanics are that 1) the MF arcs maintain similar shape
during inspiration, 2) the MTJ always has the same location
relative to a point on the CT that lies on the midplane and in the same
- plane, 3) slope and curvature are continuous at the
MTJ, and 4) points on the CT that are on the midplane remain on the midplane at all lung volumes during inspiration. If these assumptions are valid, a simple geometrical analysis shows that the
centers of the circles associated with an MF-CT arc, a point on the CT
located on the midplane, and the MTJ all follow circular trajectories
(of curvature equal to that of the MF arc) during inspiration. Circular
trajectories, however, are inconsistent with the physiological data; it
has already been shown that the trajectories of the MTJ and midplane
are highly linear (R2 > 0.96 for both and
for both postures) and parallel during inspiration.
Moreover, if CW were fixed, at least one of the key assumptions stated
above must be violated (Fig. 8). Although
either would allow the other key assumptions to remain valid,
discontinuities of slope in the CT (scenario 1)
or at the MTJ (scenario 2) are unlikely. Visual inspection
of the data suggests continuity of slope in the CT and continuity of
slope at the MTJ in the direction of MFs. To keep all other key
assumptions valid, curvature at MTJ could be made discontinuous.
However, continuity of stresses and of stiffnesses at MTJ may be
incompatible with a discontinuity of curvature at MTJ, making
scenario 3 unlikely as well. In scenario 4, one
can see that, if CW is fixed, all other key assumptions can be upheld
only if the curvature of the MF and CT increases as the muscle
contracts. More precisely, an absence of CW displacement would result
in a minimum 15% increase in diaphragm curvature in the physiological
range, assuming that no other geometrical or mechanical parameters
cause any change in curvature. However, Boriek et al. observed an 11%
decrease in curvature in the physiological range. Lateral displacement
of CW is more compatible with these observations of diaphragm
curvature. Scenario 4 is thus unlikely to be the case. In
short, if CW is fixed and if diaphragm curvature is maintained, then at
least one of the previously stated key assumptions cannot be true.
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Fig. 8.
Possible configurations of the MF arcs in the fixed CW
model. At lung volume increase, with CW fixed, either (1) slope is
discontinuous on the CT at the midline, (2) slope is discontinuous at
the MTJ, (3) curvature is discontinuous at MTJ, or (4) the curvature of
the fibers is nonconstant. The left and right fiber arcs are rotated so
as to appear to be in the same plane. Note that CT is drawn as an arc
of constant radius (R) because its shape, although undetermined by our
data, is presumed to be independent of lung volume.
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The model of diaphragm kinematics developed by Boriek et al.
(3) makes the simplifying assumption that, with a fixed
CW, a circular trajectory of MTJ can be approximated by a straight line. This is a reasonable assumption if the angle subtended by the MF
and CT from CW to midplane is ~90° and if the amount of muscle
shortening from FRC to TLC is small. Our model does not make these
assumptions, because this angle approaches 90° only as the lung
volume approaches TLC, and the MF at TLC has shortened to 65% of its
length at FRC. Looking at the relationship between diaphragm curvature
and midplane, MTJ, and CW displacements, we see that CW must move
laterally for the trajectories of the MTJ and midplane to be
essentially linear and parallel. The observed lateral displacement of
CW supports this assertion (discussed in Fig.
9).
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Fig. 9.
Lateral () displacement of the CW vs. normalized MF
length (). The dotted lines represent the theoretical (theor) range
of as a function of muscle length, predicted by the model of
Fig. 7. Both of the model functions assume that the radius of curvature
(R) of the MF arcs is 5 cm and that the angle spanned by the MF at EE
is 69°. The upper model limit assumes a CT angle of 46°; the lower
is assumed to be 40°. Both of these CT angles are within the
physiological range. The same data used in Fig. 3 and Table 1 provide
the experimental data points shown.
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As the angle subtended by the MF and CT falls below 90° (a lung
volume just greater than TLC), medial, rather than lateral, displacement of CW would be required to maintain diaphragm shape. Although this phenomenon has not been observed physiologically, the
expected decrease in the rate of lateral CW displacement with respect
to muscle shortening as the lung volume approaches TLC has been
observed visually.
Kinematic model.
Considering a CW that moves laterally during inspiration allows us to
test the hypothesis that lateral motion of CW is essential in
maintaining diaphragm shape. To maintain curvature of the MF arcs, the
kinematic model in Fig. 7 predicts that the lateral component of CW
displacement as a function of normalized muscle length
[CW()] must be
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(1)
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where R is the radius of curvature of the MF, is
normalized MF length ( = 1 at EE), EE is the
angle spanned by the MF at EE, and is the angle spanned by the CT.
Equation 1 is found from a simple geometrical analysis of
the model in Fig. 7: R × cos(EE + 
/2) is simply the distance of CW from the midplane at
EE, whereas R × cos(EE + /2) is the distance of CW from the midplane when the MF
has contracted to times its length at EE. The difference between
these two terms is the lateral displacement of CW from its location at EE.
With the availability of both model predictions (Eq. 1) and
physiological measurements (Fig. 3, Table 1) of the lateral chest wall
displacement [CW()], it is useful to compare the lateral displacement observed with that predicted by the model to
maintain curvature in the diaphragm (Fig. 9). Because the theoretical lateral displacement that would be necessary depends on the radius of
the MF-CT arc and on the angles subtended by the MF and CT, a
reasonable range of expected displacement can be generated through the
observed physiological range. As per the observations of an earlier
study (3), we chose 5 cm to be the radius of the MF-CT arc
and 69° to be the angle subtended by the MF at EE. The angle subtended by the CT, which is assumed to be constant in our model, was
found to vary between 36° and 53° in that study. Consequently, we
constructed a range of expected lateral displacement of CW based on a
6° range of freedom in the length of the CT: 40-46°. The
physiological data fit very well inside the theoretical envelope of
displacement that is predicted by the model to maintain curvature.
Although the marker data from both postures support the model, the
supine posture produces markedly less consistent results than the
prone. Figure 9 shows that the lateral displacement of CW in the supine
posture is less than that necessary to maintain diaphragm curvature.
Moreover, the low R2 values for
CW() in the supine posture
(R2 = 0.218) confirm that for the supine
posture, lateral CW displacement is not predicted well by normalized
muscle length. Thus our model of midcostal diaphragm kinematics
describes more accurately the physiological prone posture than the
unphysiological supine posture.
The same comparison can be made between model predictions and
physiological measurements of the caudal displacement of the MTJ with
respect to CW as a function of muscle length
[MTJ() CW()]. To
maintain diaphragm curvature, our kinematic model predicts that the
caudal displacement of MTJ with respect to CW must be
![&Dgr;MTJ<SUB><IT>&zgr;</IT></SUB>(<IT>&ggr;</IT>)<IT>−&Dgr;CW<SUB>&zgr;</SUB></IT>(<IT>&ggr;</IT>)<IT>=R</IT>[sin(<IT>&thgr;</IT><SUB>EE</SUB><IT>+&phgr;−&pgr;/2</IT>)<IT>−</IT>sin(<IT>&ggr;&thgr;</IT><SUB>EE</SUB><IT>+&phgr;−&pgr;/2</IT>)]](/content/vol280/issue2/fulltext/R588/img003.gif)
|
(2)
|
Equation 2 is derived using a similar geometrical
analysis as that used in deriving Eq. 1. Making similar
assumptions to those that were made in Fig. 9, Fig.
10 compares the caudal displacement of
MTJ that is predicted by the model to that which was observed. The
observed caudal displacement is only slightly greater than that
predicted by the model. Thus, when Figs. 9 and 10 are considered, the
kinematic model is able to predict both the lateral displacement of CW
and the caudal displacement of the MTJ based on the assumption that
diaphragm shape is maintained. We believe that the data confirm, to a
great extent, our hypothesis that lateral displacement of CW is crucial
in limiting changes in shape of the active midcostal diaphragm.
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|
Fig. 10.
Cranio-caudal () MTJ displacement of the relative to
the CW vs. . The dotted line is the theoretical caudal displacement
predicted by the model of Fig. 7. The solid line is a best-fit line to
the experimental data points. The model function assumes that the R of
the MF arcs is 5 cm, that the angle spanned by the MF at EE is 69°,
and that the angle spanned by the CT is 43°. A theoretical range of
displacement has not been shown as in Fig. 9, because the upper and
lower limits would be essentially superimposed on each other. The same
data used in Figs. 2 and 3 and in Tables 1 and 2 provide the
experimental data points shown.
|
|
Perspectives
Diaphragm activation has an inspiratory effect on the rib
cage by increasing abdominal pressure, thereby leading to lateral displacement of the CW in the midcostal region of the diaphragm. This
study has shown that such displacement of the CW, in turn, acts to
limit changes in shape of the active diaphragm. We have developed a
kinematic model of the canine midcostal diaphragm that demonstrates the
effect of chest wall displacement on diaphragm curvature. The data have
confirmed our hypothesis that significant lateral displacement of CW
occurs and is an essential mechanism by which changes in the shape of
the midcostal diaphragm are limited. The kinematic model includes the
lateral and caudal motion of CW during inspiration and describes these
displacements as functions of MF length. These functions are based on
physiological data that approximate well the lateral displacements
predicted by the kinematic model that would be necessary to maintain
curvature in the MF arcs. Model predictions of the motion of the MTJ
caudally and parallel to the midplane during inspiration are also
consistent with the physiological data.
It would also be interesting to examine whether the kinematic
model holds for the dorsal region of the costal diaphragm as well as
the midcostal. Because our markers were placed only on the midcostal
diaphragm, they do not reveal the kinematics of the dorsal region.
Because the dorsal region is closer to the crural diaphragm, the
kinematics of the dorsal region might be influenced by the mechanics
and activation of the crural diaphragm. For example, the crural is
stimulated earlier than the costal diaphragm, and the dorsal costal
diaphragm might be more affected by that activation. Further study of
the dorsal region could reveal whether CW expansion is a primary
mechanism by which shape is maintained in the dorsal costal diaphragm
as well.
Although this study has developed a kinematic model of the
diaphragm that is valid both for quiet, spontaneous breathing and for
inspiratory efforts against an occluded airway, it is difficult to
speculate on the applicability of this model to exercise conditions because of different levels of activation of the diaphragm and of the
intercostal muscles. It would therefore be useful to test the
extensibility of this model to exercise conditions with data gathered
from exercising dogs.
| |
ACKNOWLEDGEMENTS |
The authors thank Anjelica Gonzales for help with the data analysis.
| |
FOOTNOTES |
This work was supported by National Heart, Lung, and Blood Institute
Grant HL-46230.
Address for reprint requests and other correspondence: A. M. Boriek, Pulmonary and Critical Care Section, Suite 520B, One Baylor
Plaza, Houston, TX 77030 (E-mail: boriek{at}bcm.tmc.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 14 February 2000; accepted in final form 28 September 2000.
| |
REFERENCES |
1.
Boriek, AM,
Liu S,
and
Rodarte JR.
Costal diaphragm curvature in the dog.
J Appl Physiol
75:
527-533,
1993[Abstract/Free Full Text].
2.
Boriek, AM,
and
Rodarte JR.
Effects of transverse fiber stiffness and central tendon on displacement and shape of a simple diaphragm model.
J Appl Physiol
82:
1626-1636,
1997[Abstract/Free Full Text].
3.
Boriek, AM,
Rodarte JR,
and
Wilson TA.
Kinematics and mechanics of the midcostal diaphragm of the dog.
J Appl Physiol
83:
1068-1075,
1997[Abstract/Free Full Text].
4.
Boriek, AM,
Wilson TA,
and
Rodarte JR.
Displacements and strains in the costal diaphragm of the dog.
J Appl Physiol
76:
223-229,
1994[Abstract/Free Full Text].
5.
Hubmayr, RD,
Sprung J,
and
Nelson S.
Determinants of transdiaphragmatic pressure in dogs.
J Appl Physiol
69:
2050-2056,
1990[Abstract/Free Full Text].
6.
Kim, MJ,
Druz WS,
Danou J,
Machnack W,
and
Sharp JT.
Mechanics of the canine diaphragm.
J Appl Physiol
41:
369-382,
1976[Abstract/Free Full Text].
7.
Road, J,
Newman S,
Derenne JP,
and
Grassino A.
In vivo length-force relationship of canine diaphragm.
J Appl Physiol
60:
63-70,
1986[Abstract/Free Full Text].
8.
Whitelaw, WA,
Hajdo LE,
and
Wallace JA.
Relationship among pressure, tension, and shape of the diaphragm.
J Appl Physiol
55:
1899-1905,
1983[Abstract/Free Full Text].
Am J Physiol Regul Integr Comp Physiol 280(2):R588-R597
0363-6119/01 $5.00
Copyright © 2001 the American Physiological Society
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ABSTRACT
INTRODUCTION
![<IT>−</IT>cos(<IT>&thgr;</IT><SUB>EE</SUB><IT>+&phgr;−&pgr;/2</IT>)]](/content/vol280/issue2/fulltext/R588/img002.gif)
