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A kinematic model of swallowing in Aplysia californica based on radula/odontophore kinematics and in vivo magnetic resonance images
1 Department of Biomedical Engineering, Case Western Reserve University,
Cleveland, OH 44106-7080, USA
2 Department of Biology Case Western Reserve University, Cleveland, OH
44106-7080, USA
3 Department of Neurosciences, Case Western Reserve University, Cleveland,
OH 44106-7080, USA
4 MR Systems Department, G. E. Medical Systems Israel Ltd, Keren Hayesod
Street, PO Box 2071, Tirat Carmel 39120, Israel
* Author for correspondence at address 2 (e-mail: hjc{at}po.cwru.edu)
Accepted 3 July 2002
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Summary |
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 TOP Summary IntroductionMaterials and methodsResultsDiscussionReferences |
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Key words: feeding, behaviour, biomechanics, kinematics, mollusc, muscular hydrostat, Aplysia californica
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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). The
molluscan feeding apparatus is characterized by a rasping surface (the radula)
and an underlying muscular structure (the odontophore) that is often
associated with a cartilaginous structure (bolsters or rotellae;
Starmühlner, 1956). A
variety of different hypotheses have been proposed to account for the function
of these structures. For example, investigators have proposed that the radular
surface and underlying odontophore might act like a pulley, a block and
tackle, a rasp or a conveyer belt (Eales,
1921; Howells,
1942; Smith,
1988). These hypotheses have been based on observations of the
parts of the structure visible during portions of the feeding cycle,
histological characterizations of the anatomy and stimulation of individual
muscles. For example, careful kinematic analysis of radular kinetics during
grazing in Helisoma trivolvis indicated that the radula slides over
the underlying cartilage, which is independently accelerated during each
feeding stroke, supporting a moving conveyor belt hypothesis
(Smith, 1988).
Analysis of the radula and odontophore within the buccal mass is
complicated by the absence of hard skeletal elements and discrete joints that
make musculo-skeletal systems tractable to mechanical analysis. Molluscan
feeding structures are composed entirely of muscle and cartilage, and muscle
acts both to generate forces and to provide skeletal support. Thus, they are
examples of a broader class of structures, muscular hydrostats, that are
exemplified by tongues, trunks and tentacles
(Kier and Smith, 1985).
Because these structures have many degrees of freedom and are thus capable of
complex and flexible movements, understanding their biomechanical properties
is likely to be essential for a deeper understanding of their neural control.
Moreover, the great flexibility of these structures allows them to be utilized
for multiple different behavioral functions (e.g. the human tongue is used
both for feeding and for talking), and thus the neural architectures
controlling these devices are also of special interest for understanding the
dynamics of multifunctionality.
We have focused on analyzing the biomechanics and neural control of feeding
in the marine mollusc Aplysia californica. Aplysia is a generalist
herbivore that feeds on a variety of red, brown and green seaweeds whose
shapes, toughness and texture vary significantly
(Carefoot, 1967;
Howells, 1942;
Pennings, 1990). The feeding
behavior of Aplysia is under the control of motivational variables
(Kupfermann, 1974) and is
subject to associative learning (Chiel and
Susswein, 1993; Susswein et
al., 1986). The neural control of the feeding apparatus in
Aplysia has been intensively studied. Sensory neurons responsive to
chemical or mechanical stimuli that induce consummatory feeding responses have
been identified (Miller et al.,
1994; Rosen et al.,
1979,
1982,
2000a,b),
as have motor neurons for the major muscles of the feeding apparatus
(Church et al., 1991;
Church and Lloyd, 1994;
Gardner, 1993). Neural
correlates that distinguish ingestion from rejection have been defined
(Cropper et al., 1990a; Morton
and Chiel,
1993a,b)
and have been used to identify interneurons responsible for flexibly shifting
the timing and intensity of activation of motor neuronal pools so that
ingestive or egestive behavior can be generated under appropriate conditions
(Hurwitz et al., 1997;
Jing and Weiss, 2001).
Interneurons responsive to mechanical load have been shown to cause the switch
from biting to swallowing (Evans and
Cropper, 1998).
The kinematics of the buccal mass of Aplysia have also begun to be
clarified. Earlier studies clarified the functional anatomy of the intrinsic
muscles (labelled `I' followed by a number) and extrinsic muscles (labelled
`E' followed by a number; Howells,
1942). In the present paper,
Fig. 21 provides a schematic
view of the buccal mass musculature, and
Fig. 19 provides a schematic
view of the muscles of the radula/odontophore proper. A series of kinematic
models of the entire buccal mass has been constructed (Drushel et al.,
1998,
2002). These models have
provided an increasingly accurate view of the inner workings of the buccal
mass, but may not have completely captured the three-dimensional shape of the
radula/odontophore. A previous attempt to capture the three-dimensional shapes
of the radula/odontophore throughout the feeding cycle
(Drushel et al., 2002) used
two different approaches. In one approach, the radular halves could move
relative to one another and to the radular stalk, creating a three-dimensional
shape. This model was referred to as a radular-centric model. In the other
approach, the mid-sagittal shape of the odontophore was constrained to be
identical to that observed in mid-sagittal magnetic resonance images (MRIs),
and the remainder of the three-dimensional shape of the odontophore was
determined from the volume of the buccal mass and assumptions about its
medio-lateral width. This model was referred to as an odontophore-centric
model.
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If the full three-dimensional shapes of the buccal mass and its constituent
muscles could be simultaneously measured in intact, behaving animals, it would
be possible to develop a complete kinematic description of the musculature.
Since this is not currently technically feasible, we have developed a
technique for obtaining high-temporal- and spatial-resolution planar images of
feeding in intact animals using magnetic resonance imaging (MRI). In addition,
by inducing feeding-like movements in isolated odontophores in response to
pharmacological agents (Drushel et al.,
1998; Susswein et al.,
1996), it was possible to analyze the kinematics of isolated
radula/odontophores in order to derive a set of kinematic relationships for
its three-dimensional deformations. By extracting parameters from mid-sagittal
MRIs of the radula/odontophore in intact, behaving animals and using them as
inputs to a kinematic model based on these kinematic relationships, it was
possible to reconstruct the three-dimensional shape of the radula/odontophore
throughout the feeding cycle. By combining these odontophore model shapes with
a kinematic model of the surrounding musculature, we generated a new
odontophore-centric three-dimensional kinematic model of the buccal mass.
After validating the overall model, we used it to describe the kinematics of
buccal muscles and buccal mass components during swallowing, and compared
these predictions with actual measurements. The model generated several
testable hypotheses about the context-dependent function of components of the
buccal mass that have significant implications for its neural control.
Portions of this work have appeared in preliminary form
(Neustadter et al., 2001).
As adjuncts to the text, we provide digital movies (in QuickTime format) of the MRIs of swallowing in Aplysia californica used for the model presented in this paper, movies of the model construction and movies of the model output. The movie entitled `3_15_39highres.mov' shows interleaved sagittal, coronal and axial images of the buccal mass during swallowing from sequence 7732-S3, frames 15-39. The movies entitled `ModelProcess.mov', `ModelProcess2.mov', `ModelProcess3.mov', `ModelProcess4.mov' and `ModelProcess5.mov' illustrate the process by which the three-dimensional kinematic model of the odontophore and the buccal mass is constructed, and will clarify the Materials and methods section. The movies entitled `16-39ModelSideView', `16-39ModelTopView.mov' and `16-39ModelFrontView.mov' show side, top and front orthogonal projections of the kinematic model of the buccal mass for sequence 7732-S3, frames 16-39. These movies will clarify the Results section.
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Materials and methods |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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Measurements of the kinematics of the radula/odontophore
To create a complete three-dimensional model of the changing shapes of the
radula/odontophore during a feeding cycle, we needed to determine kinematic
relationships that would allow us to infer the overall shape of the structure
from planar mid-sagittal MRIs of the structure during feeding. We therefore
videotaped and analyzed the relationships between three-dimensional anatomical
features seen in multiple planar views of isolated, intact radula/odontophores
during spontaneous and drug-induced feeding-like movements. Aplysia
californica Cooper (160-303 g, obtained from Marinus, Long Beach, CA,
USA) (N=8) were anesthetized by gradually lowering their body
temperature to 4°C using a dissecting tray filled with ice and placing
them in a freezer for 30 min. For some studies, animals were anesthetized
using magnesium chloride (isotonic 333 mmol l-1 MgCl2
equal to half their body mass). The buccal mass was dissected out along with
the cerebral and buccal ganglia. The buccal mass was then placed in a dish
containing artificial seawater (Instant Ocean, Mentor, OH, USA) at room
temperature. The dorsal surface of the buccal mass was cut in an
antero-posterior direction along the mid-sagittal line back to the dorsal
surface of the esophagus. Much of the I1/I3 tissue on either side of the
ventral surface of the radula/odontophore was dissected away so that the base
of the radula/odontophore was exposed.
Multiple planar views of the radula/odontophore were obtained simultaneously by mounting two mirrors at 45° to the camera axis, providing three perpendicular views of the preparation that were captured in a single video image. The odontophore was mounted below the mirror that provided a top view and to the left of the mirror that provided a front view. The odontophore itself was oriented to provide the video camera with a side view. A light was shone onto the preparation from above, so that the odontophore's widest medio-lateral extent could be determined during movement by examining the line of shadow that it cast. In one preparation, the anterior edge of the radula/odontophore was mounted on a vertical pin using silk sutures so that the radula/odontophore would have a fixed frame of reference (Fig. 1). Digital NTSC video (Canon ZR10, Canon Inc., Jamesburg, NJ, USA; 30 frames s-1) was used to record the movements of the preparation.
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Feeding-like movements were obtained in several ways. As the buccal mass
recovered from anesthesia, vigorous spontaneous movements were observed.
Crystals of carbachol or dopamine hydrochloride (C-4382 or H-8502,
respectively; Sigma, St Louis, MO, USA) were placed on the cerebral ganglion,
inducing rhythmic movements (Drushel et
al., 1998; Susswein et al.,
1996).
Kinematic measurements indicated that several features of the
radula/odontophore contributed significantly to the distribution of its volume
and should therefore be represented in the three-dimensional model. Moreover,
the planar views indicated that the positions and dimensions of these features
could be deduced from a mid-sagittal slice (see below). In particular, we
identified a wedge-shaped structure that appears to be filled with fluid and
is anterior to the I6 muscle, which we refer to as the prow of the odontophore
(Fig. 1) (see also
fig. 6 in
Neustadter et al., 2002). We
also recognized that the anterior surface of the radula does not curve
smoothly, but forms a ridge as a result of the upward protrusion of the I4
muscles (Fig. 1) (see also
fig. 6 in
Neustadter et al., 2002). We
tracked the movements of these features, as well as monitoring the changing
position of the shadow, i.e. the line of widest medio-lateral extent, during
the feeding-like movements of the radula/odontophore (see Results for
measurements and below for a description of how the kinematic relationships
were used to deduce rules for the construction of the three-dimensional model
of the radula/odontophore).
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Measurements of the volume of the radula/odontophore
Given linear measurements from the mid-sagittal plane, and assuming that
the radula/odontophore is isovolumetric throughout the feeding cycle, it is
possible to use a scaled estimate of the volume to determine the medio-lateral
width of the radula/odontophore. We therefore measured the resting volume of
the odontophore. The apparatus used to make these measurements consisted of a
60 ml syringe clamped in an upright position and connected via a tube
to a 0.2 ml glass pipette, which was also clamped in an upright position to
approximately the same height. The pipette was used to provide a narrow water
column in which small changes in water level could be accurately recorded.
Changes in the water level were determined by measuring the height of the
meniscus of the fluid in the pipette through a microscope whose eyepiece was
equipped with a graduated reticle. To minimize surface tension, which
interfered with the free movement of the meniscus, the apparatus was soaked in
a solution containing soap (Alconox Detergent Powder; Alconox Inc., New York,
NY, USA) for at least 24 h prior to measurements, after which it was rinsed
and filled with artificial seawater. The apparatus was calibrated by adding
known volumes of water (using an Eppendorf pipette to deliver precise 0.5 ml
samples to the apparatus) and recording the changes in the height of the
meniscus. The precision of the measurements was ±0.05 ml, and the
volumes of the odontophores ranged from 0.4 to 1.5 ml, so that the largest
error in measurement of the smallest odontophore was approximately 12%.
As described in previous work (fig.
3 in Neustadter et al.,
2002), we have used the internal radular stalk width as a
reference length that normalizes lengths and volumes among animals so as to
combine measurements from isolated odontophores of different sizes and
mid-sagittal MRIs. The volume of the odontophore was therefore normalized to
units of (radular stalk width)3, which we refer to as
RSW3. From measurements performed on five animals ranging in mass
from 65 g to 335 g, the mean odontophore volume including the prow and the
stalk was computed to be 7.5±0.6 RSW3 (mean ± S.D.,
N=5).
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Kinematic model of the buccal mass
The kinematic model consists of the following components: (i) a model of
the radula/odontophore, whose three-dimensional shape is based on the
kinematic relationships deduced from the studies described above; (ii) a model
of the surrounding I3 musculature, based on a modified version of a previous
model of these structures (Drushel et al.,
1998,
2002); and (iii) an iterative
algorithm that positions the radula/odontophore relative to the I3 model
muscles so as to best fit the mid-sagittal outline of the buccal mass. We will
describe each of these components in turn.
Radula/odontophore model
The three-dimensional shape of the radula/odontophore includes the prow and
the radular ridge. The `radular cleft', i.e. the space between the halves of
the radula when it is open, is not included in the model, because it was not
possible to identify a measurable parameter on the mid-sagittal MRI that could
be used to deduce its medio-lateral and dorso-ventral extents. Potential
errors that this introduces in the volume of the radula/odontophore are
considered in the Discussion. The model also does not include the narrow ridge
that the radular surface forms during and after the peak of retraction (which
we refer to as the radular `pinch'), again because of the absence of a
measurable parameter to indicate its extent on the mid-sagittal MRI.
Parameters
Fixed parameters for the model were measured from
high-spatial-resolution MRIs of anaesthetized animals and isolated buccal
masses, high-spatial-resolution photographs of isolated odontophores and
digital video recordings of radula/odontophore kinematics, as described above.
The fixed parameters are (i) the overall volume of the odontophore, (ii) the
volume of the radular stalk, (iii) the spline parameters describing the
vertical and horizontal cross sectional shapes of the odontophore, (iv) the
shape of the prow, (v) the volume of the prow and (vi) the parameters defining
the shape of the ridge (see Fig.
2; Table 1; see
also fig. 3C,D in
Neustadter et al., 2002). The
spline parameters are used to describe curves using two control points to
define a smooth curve between two endpoints
(Press et al., 1988) (see
legend to Fig. 2).
Parameters for each model frame were measured from mid-sagittal
high-temporal-resolution MRIs. The parameters were (i) the outline of the
odontophore, including the prow, (ii) the line separating the prow from the
odontophore corresponding to the anterior margin of I6, (iii) the point above
which to search for the tip of the prow and (iv) the position and orientation
of the stalk. The shape of the stalk was based on an outline of the stalk from
high-spatial-resolution MRIs and was fixed (see
fig. 4 in
Neustadter et al., 2002). The
stalk outline was then scaled for each animal so that it fitted onto the stalk
in the image. The following measurements were also made: (v) the outline of
the entire buccal mass (including the radular stalk, odontophore and the I3
musculature, but excluding pharyngeal tissue); (vi) the `lateral groove' (the
most posterior part of the I3 musculature); (vii) the `hinge' (the point of
attachment of the ventral radula/odontophore and the most posterior part of
the I3 musculature); (viii) the `line of the jaws' (the location of the most
anterior part of the I3 musculature); (ix) an upper limit line that indicated
the inner border of the dorsal section of I3; and (x) a lower limit line that
indicated the inner border of the ventral section of I3. Extraction of
parameters i, ii, iv, v, vi and viii is illustrated in
fig. 4 of Neustadter et al.
(2002).
Construction of the odontophore
The model creates a three-dimensional mesh that represents the shape of the
odontophore. If the odontophore were spherical, one could construct its shape
using vertices lying on a number of parallel circles. The diameter of the
central circle would be the diameter of the sphere, and the diameters of the
other circles would decrease as the circles were further from the center,
reaching zero at the front and back of the sphere. To provide an approximately
uniform distribution of the vertices on the surface of the sphere, the circles
should be spaced in an antero-posterior density corresponding to a cosine
function, i.e. more closely spaced at the front and back of the sphere than in
the middle.
Because the actual odontophore has a more complex shape, the curves used to define its vertices are not circles but more complex closed convex curves constructed of four spline quadrants. The spline parameters defining the shape of each quadrant are based on projections of radula/odontophores (Fig. 2B,E). The four spline quadrants that compose each curve connect at four anchor points (Fig. 3C). The dorsal and ventral anchor points are defined by the intersection of the plane of the curve with the outline of the odontophore extracted from the mid-sagittal high-temporal-resolution MRIs (Fig. 3A,C). The medio-lateral anchor points are defined by the intersection of the plane of the curve with a curve defining the medio-lateral width (described in the next section; Fig. 3B,C). The spacing between the curves in the antero-posterior direction is cosinusoidal (more closely spaced at the anterior and posterior edges, less closely spaced at the center) to provide approximately uniform coverage of the surface of the odontophore (Fig. 4C).
Determining the angles of the planes of the closed curves that define
the odontophore mesh
The maximum width of each of the closed curves is defined by its
intersection with a curve in a plane that cuts through the structure along its
widest medio-lateral extent (Fig.
3B). In the previous odontophore-centric model of the
radula/odontophore (Drushel et al.,
2002), this plane was assumed to be the plane that passed through
the anterior and posterior extremes of the mid-sagittal cross section of the
odontophore. As a consequence, a series of vertical closed curves could be
placed parallel to one another along the line from the anterior extreme to the
posterior extreme to include the entire volume of the odontophore
(Fig. 4A). Studies of the
shadow line on the isolated radula/odontophore during feeding-like movements
(see below; see also Fig. 7)
demonstrated that the line of widest medio-lateral extent did not necessarily
intersect the anterior and posterior extremes of the mid-sagittal cross
section of the odontophore. Consequently, if the closed curves were placed
parallel to one another and their width was defined by their intersections
with the line of widest extent, a portion of the volume of the odontophore
would extend past the ends of the line of widest extent and would, therefore,
have no defined width (Fig.
4B). To solve this problem, we computed the tangents to the
mid-sagittal outline at the posterior and anterior ends of the line of widest
extent and set the closed curves at angles that continuously changed from the
posterior tangent to the anterior tangent value. This guaranteed that the
entire volume was represented by curves that intersected the line of widest
extent (Fig. 4C).
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Determining the line of widest extent
The position and angle of the line of widest extent were derived using a
kinematic relationship based on observations of isolated, moving
radula/odontophores. The line of widest extent, which was observed as a shadow
line on the moving odontophores, was found to have a fixed angle (44°)
relative to the line connecting the top of the radular surface and the tip of
the prow and to pass through the tip of the prow (see below; see also
Fig. 7). This relationship
allowed us to deduce the angle of the line of widest extent, which cannot be
directly observed in the mid-sagittal MRIs, from the angle of the line
connecting the top of the radular surface and the tip of the prow, which can
be directly measured in the MRIs.
To calculate the angle of the line of widest extent in the model, the
location of the tip of the prow, the location of the top of the radular
surface and the angle of the line connecting them must be determined.
Anatomically, the tip of the prow is the anteriormost end of the radular
surface, and there is therefore a large change in curvature at that point
along the mid-sagittal outline of the prow
(Fig. 3D). To locate the tip of
the prow objectively, we therefore implemented an algorithm that identifies
the point of sharpest curvature along the top portion of the mid-sagittal
outline of the prow. The algorithm selected the point at which the sum of the
2 goodness-of-fit errors for linear fits to the portions of
the curve above and below the point was minimal. Because the tip and the
bottom of the prow were sutured to a vertical pin, this vertical line defined
the reference frame in which the top of the radular surface was measured in
the moving isolated odontophores. Consequently, before determining the
location of the top of the radular surface for the model, the mid-sagittal
cross section was rotated so that the line connecting the tip of the prow and
the bottom of the prow was vertical (Fig.
3E). After applying this rotation, we identified the topmost point
of the mid-sagittal cross section, which was the top of the radular surface.
We then defined the line of widest extent to be the line that passed through
the tip of the prow and was 44° counterclockwise from the line connecting
the top of the radular surface and the tip of the prow (relative to the
orientation shown in Fig.
3E).
Construction of the prow
Anatomical studies of the prow indicated that its volume could be treated
as constant, that its posterior margin formed a plane with the I6 muscle and
that the medio-lateral profile of its anterior edge could be approximated by a
Gaussian curve (see Fig. 6C in
Neustadter et al., 2002). The
Gaussian function used to determine the medio-lateral width x of the
prow in terms of its antero-posterior location was:
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Two parameters for the prow were extracted from each mid-sagittal MRI: (i)
the mid-sagittal anterior margin of the prow and (ii) the line along which it
meets I6, which we refer to as the `prow seam' (see
fig. 4B,C in
Neustadter et al., 2002; the
anterior margin of the odontophore corresponds to the prow seam). The
antero-posterior position of the prow seam is later iteratively adjusted to
achieve the correct prow volume, but its angle remains unchanged. The entire
three-dimensional mesh is constructed in the reference frame in which the prow
seam is vertical. To best approximate the actual anatomical shape of the prow
(see Fig. 6 in
Neustadter et al., 2002), the
dorsal half of each curve defining the volume of the prow (i.e. the part of
the curve above the line of widest extent) was vertical (i.e. parallel to the
prow seam) and the ventral half (i.e. the part of the curve below the line of
widest extent) curved posteriorly to meet the other curves at the base of the
prow seam (Fig. 3F). The
curvature of the ventral half is such that its position is a fixed percentage
of the distance between the anterior margin of the prow and the prow seam. The
shapes of the dorsal and ventral parts of each curve were defined by the same
spline parameters used to define the curves of the rest of the odontophore,
and the maximum width at the intersection with the line of widest extent was
defined by the Gaussian approximation of the medio-lateral profile of the
prow. The width of the Gaussian curve at the prow seam was defined as 0.77
RSW, based on anatomical observations. To achieve the known fixed volume of
the prow, the mesh representing the prow was iteratively constructed with the
prow seam being moved anteriorly or posteriorly.
Construction of the ridge
Observations of the kinematics of isolated radula/odontophores indicated
that it was necessary to include a dorsal ridge. The spline curve used to
approximate the shape of the dorsal half of the odontophore does not
accurately represent the shape of the dorsal surface throughout the movements
of the isolated odontophore (in Fig.
2E, note the discrepancy between the location of the spline curve
and the dorsal surface of the odontophore). During a portion of the movement
cycle, a ridge projects above this shape. To determine the timing and extent
of ridge protrusion during a movement cycle in the isolated
radula/odontophore, we used the top view of the radula to locate the posterior
and anterior margins of the ridge, and identified these locations in the side
view (Fig. 5A). On the basis of
empirical measurements, we found that a circular arc (radius 1.23 RSW, arc
angle 100°) could be fitted to the dorsal radular surface anterior and
posterior to the ridge (Fig.
5B). This arc was therefore fitted to the anterior portion of the
odontophore outlines measured on the mid-sagittal MRIs and used to define the
ridge for the model. The arc was fitted to 0.25 times the distance between the
tip of the prow and the posterior of the odontophore. The arc was continued
posteriorly by a straight line tangential to the posterior end of the arc
(Fig. 5B), on the basis of
empirical observations. A protrusion above this arc indicated the presence of
a ridge. If a ridge was observed in the mid-sagittal section of the MRI, we
constructed the medio-lateral shape of the ridge (based on empirical
observations of the ridge in vitro) by constructing a trapezoid whose
ventral width was 0.46 RSW less than the maximum width of the odontophore, and
whose dorsal width was 0.4 times the ventral width. The radius of curvature of
the corner of the trapezoid in the medio-lateral dimension was 0.2 RSW
(Fig. 5C, top). For each curve
in the region of the protrusion, the dorso-ventral height of the ridge was
defined by the height of the protrusion, and the medio-lateral width of the
ridge was defined relative to the width of the odontophore at that location.
Since the planes of the curves are not necessarily vertical
(Fig. 4C), the medio-lateral
odontophore width used to determine the ridge width was the maximum of two
widths: (i) the width at the intersection of the curve with the line of widest
extent or (ii) the maximum odontophore width at the horizontal location of the
top of the curve. If the width of the odontophore was less than 1.0 RSW, no
ridge was constructed. Otherwise, a trapezoid representing the ridge was
constructed by relocating the vertices of the curve defining the dorsal edge
of the odontophore mesh (Fig.
5C, bottom).
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Total volume of the odontophore
Although it was possible to determine the location of the line of widest
extent from a mid-sagittal MRI using the kinematic relationships described
above, and the shape of the curve in the plane of widest extent was determined
by the spline parameters measured from the high-spatial-resolution MRIs of the
anesthetized buccal mass, the actual medio-lateral width of the
radula/odontophore could not be determined from a mid-sagittal image. Given
the assumption that the radula/odontophore is isovolumetric and given the
measured volume of the odontophore, the medio-lateral width of the
three-dimensional mesh was iteratively adjusted, and the odontophore mesh was
reconstructed until the correct volume was achieved. Determining the correct
volume required a measure of the volume of the radular stalk and the extent to
which it overlapped the volume of the odontophore. The volume of the radular
stalk was calculated from a measurement of the radular stalk width and a
three-dimensional reconstruction of the radular stalk from
high-spatial-resolution MRI (see fig.
3C in Neustadter et al.,
2002; the value is 0.69 RSW3). The volume of the
radular stalk was assumed to be approximately constant, on the basis of the
approximately constant area of its cross section in the mid-sagittal MRIs. The
position and angle of the radular stalk relative to the odontophore were
extracted from the mid-sagittal MRIs. If the radular stalk protruded through
the ventral side of the odontophore, as it does during retraction, the
protruding portion of the stalk was not included in the outline of the
odontophore. To calculate the total volume of the radula/odontophore, it was
therefore necessary for the model to calculate the sum of the volumes of the
odontophore and the radular stalk and then to subtract the overlapping volume.
If the total volume of the odontophore was incorrect, the model iteratively
adjusted the medio-lateral width at the line of widest extent and repeated the
construction of the three-dimensional odontophore mesh until the volume fell
within a pre-defined tolerance (±0.1 RSW3 of 7.5
RSW3).
Modifications to the kinematic model of the I3 musculature
Once the three-dimensional mesh of the odontophore had been constructed, a
model of the I3 musculature was constructed around it based on the previously
published kinematic model (Drushel et al.,
1998,
2002), which approximates the
I3 muscle as a number of distinct rings
(Fig. 6A) whose parameters were
estimated by trial and error. Because the high-spatial resolution MRIs
provided more information about the actual shape of the I3 musculature than
had been previously observed, we extracted additional parameters and modified
the I3 model so that we could make use of these parameters to produce a better
match between the model and the in vivo data.
The shape and size of each model I3 ring are defined by five parameters (Fig. 6A): r, the radius of the half-circular cross section of the outer half-ring at the top and bottom and of the inner half-ring surrounding the lumen; the thickness of each model ring is 2r; a, half the maximum width of the lumen; q, the width added between the outer and inner half-rings so that the medio-lateral width of the model ring matches the medio-lateral width of the I3 muscle at that location; b1, the height of the lumen above its maximum width; and b2, the height of the lumen below its maximum width. In the model I3 rings, the lumen is centered relative to the dorso-ventral length of the ring, but the widest point of the lumen and of the I3 ring need not be at the midpoint of the dorso-ventral length (i.e. b1 and b2 could have different values). Each ring can have a unique set of parameter values.
High-spatial-resolution MRI of anesthetized buccal masses indicated that the lumen was not necessarily centered along the dorso-ventral length and that the maximum medio-lateral width of the lumen was not necessarily coincident with the maximum medio-lateral width of the muscle (Fig. 6B). As a consequence, the following procedure was used to extract the five model parameters: (i) parameter a was determined by measuring the maximum width of the lumen and dividing by two; (ii) parameters b1 and b2 were determined by measuring the height of the lumen above and below its maximum width, respectively; (iii) the maximum dorso-ventral height (h) of the muscle was measured, and r was calculated as (h-b1-b2)/4, since, in the model, the total dorso-ventral height h of the I3 ring medially is 4r+b1+b2 (Fig. 6A); (iv) the maximum medio-lateral width (w) of the muscle was measured, and q was calculated as (w-2a-4r)/2 since, in the model, the total medio-lateral width w of the I3 ring at its dorso-ventral midpoint is 4r+2q+2a.
The resting volume of the I3 rings served as a constraint for each ring so that, as a ring was placed around the odontophore during the feeding cycle, the ring's parameters were adjusted iteratively to maintain its constant volume. Parameter values for the model I3 rings, and their volume, were obtained from analysis of the resting anatomy of the I1/I3/jaw muscle. High-spatial resolution MRIs generated parameters for a series of 12 1.5 mm thick antero-posterior slices through the I3 muscle. Using the techniques described in the previous paragraph, a set of parameters was extracted for each of these 12 slices. This parameter set was then converted into resting parameters for an equivalent set of six model I3 rings using the following procedure. (i) Starting at the lateral groove, and starting with the r value of the slice closest to the lateral groove, all slices that were within a distance 2r of the lateral groove were analyzed and their parameters were averaged. (ii) Since, in general, this led to a new value of r, steps (i) and (ii) were repeated until the value of r stopped changing. (iii) Starting at the anterior end of the previous ring, steps (i) and (ii) were repeated for each additional ring until the r value estimated from the anatomy, multiplied by the number of remaining model rings, gave an antero-posterior distance that would not reach the jaws (in our measurements, this occurred for the fourth ring to be calculated, which is ring 3 in Table 1). For all remaining rings, a constant r value was then used so that the total thickness of the six model I3 rings would span the distance from the lateral groove to the jaws when the buccal mass was at rest. The parameters for the I3 rings with fixed r values were assigned by averaging the parameter values extracted from the equivalent MRI slices, based on the fixed r value. (iv) Finally, since the I3 and odontophore are anatomically attached by elastic tissue which stretches, and the six I3 rings are non-elastic, the calculated parameters were scaled (by 1.04) to model units such that the model I3 spanned the distance from the jaws to the lateral groove. The resulting parameter values are listed in Table 1.
Because both the high-spatial- and high-temporal-resolution MRIs showed that the inner margins of the I3 muscle were not straight lines, the I3 model was also modified to accept dorsal and ventral limit curves that could be arbitrary polynomials rather than straight lines. This made it possible to partially represent the ability of the I3 muscle to conform smoothly to the changing internal shape of the radula/odontophore, which was clearly visible in the mid-sagittal MRIs (e.g. see Fig. 9).
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Iterative positioning algorithm
As demonstrated previously (Neustadter
et al., 2002), the midsagittal MRI provides detailed information
about the entire shape of the buccal mass during feeding. Instead of
attempting to match two idealized features of the shape, the ellipticity and
the eccentricity, as in previous models (Drushel et al.,
1998,
2002), the present model
attempted to match the entire irregular outline of the buccal mass. We
implemented an iterative placement algorithm for this purpose, which met two
constraints. First, the anterior I3 ring was required to meet the jaw line.
Second, the outer border of the I3 rings had to match the outline of the
buccal mass. The following degrees of freedom were modified if the I3 rings
did not contact the jaw line or fit within the outline of the buccal mass: (i)
the position of the hinge (i.e. the placement of the posterior I3 ring), which
primarily affected the ventral placement of the I3 rings; (ii) the angle of
the posterior ring relative to the odontophore, which primarily affected the
dorsal placement of the I3 rings; and (iii) the polynomial limit lines, which
primarily affected the match to the outline.
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Results |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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).
Kinematic relationships for the radula/odontophore
To measure the location of the maximum medio-lateral width of the
radula/odontophore throughout a feeding cycle, we examined the shadow cast by
a light above the isolated odontophore. We observed that the shadow formed a
line whose angle changed throughout the feeding-like movement cycle. We also
noted that the line connecting the tip of the prow and the top of the radula
had a fixed angle relative to the shadow line as it moved
(Fig. 7A,B; the angle between
the lines was 44±5°; mean ± S.D., N=15 measurements
from two cycles). It is therefore possible to deduce the angle of the line of
widest extent from the angle of the line connecting the tip of the prow to the
top of the radula, which can be measured in mid-sagittal MRIs.
How do the kinematics of the line connecting the tip of the prow and the top of the radula compare between isolated odontophores and odontophores within the buccal mass during an in vivo swallowing movement? We examined this question by measuring the same line on odontophores during swallowing sequences in vivo. We observed that the line connecting the tip of the prow and the top of the radula showed very similar changes in angle throughout the cycle [Fig. 7C; the timing of the different in vivo behavioral periods (t4, t1 and t2) used in this and subsequent figures is provided in the figure legend]. In turn, this suggests that inferences about the line of widest extent based on the experimentally defined kinematic relationship are likely to be valid throughout the feeding cycle in vivo.
We also characterized the kinematics of the large ridge that appears on the surface of the radula during feeding movements. After fitting a circular arc to the anterior portion of the radular surface (in a side view), the protrusion of the dorsal part of the odontophore above this arc correlated well with the ridge seen in a top view. Fig. 8A shows this relationship for several key frames for ridge protrusion in side and top views (r2=0.84, P<0.002). Thus, one can use this kinematic relationship to deduce the anterior and posterior borders of the ridge and its height from the mid-sagittal MRIs.
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How do the kinematics of the ridge observed in isolated odontophores
compare with their kinematics in radula/odontophores within the buccal mass
during swallowing in vivo? We determined the extent to which the
ridge protruded above the radular surface in isolated odontophores as they
underwent distinctive shape changes and compared them with the ridge area of
odontophores during swallowing sequences in vivo
(Fig. 8). Our measurements
in vitro indicated that the ridge was most prominent from the time
that the radula closed and the odontophore elongated dorso-ventrally (event 4,
Fig. 8B) until the time that
the radular halves opened (event 6, Fig.
8B). On the basis of our previous analysis of the mid-sagittal
in vivo kinematics (fig.
12, right side, in Neustadter
et al., 2002), this corresponds to the early retraction phase of
swallowing. Interestingly, the ridge area measured from in vivo
mid-sagittal images reaches its maximum at the onset of retraction
(Fig. 8C, border of t4
and t1 periods). In turn, this suggests that inferences about the
ridge based on the experimentally defined kinematic relationship are likely to
be valid throughout the feeding cycle.
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Model match to buccal mass kinematics
The kinematic model successfully matched the mid-sagittal outlines of the
buccal mass measured from the high-temporal-resolution MRIs
(Fig. 9). The only significant
mismatches occurred in the posterior rings of the I3 muscle, which often
extended beyond the outline of the buccal mass. This mismatch is due to an
inherent limitation of the current I3 model (see Discussion).
The most important assumption of the model was that kinematic relationships
derived from isolated radula/odontophores performing feeding-like movements
in vitro would generate valid predictions for the medio-lateral width
of the radula/odontophore throughout the feeding cycle in vivo. It
was possible to test this critical assumption because, during data
acquisition, mid-sagittal MRIs were interleaved with axial and coronal images
of the buccal mass (Neustadter et al.,
2002). By sectioning the three-dimensional model buccal mass at
the location of the corresponding coronal MR slice, it was possible to
generate coronal slices through the model that could be directly compared with
coronal MRIs (Fig. 10).
Because the coronal images were temporally interleaved between the
mid-sagittal images (see Neustadter et
al., 2002), we compared each coronal image with the model frame
based on the mid-sagittal image preceding the coronal image, with the model
frame following the coronal image and with a combination of the preceding and
following model frames, and used the best match for the comparison. A
quantitative test of the validity of the model was performed by computing the
symmetric difference (Alt et al.,
1998) between the outlines of the MR and model images as a
percentage, i.e. 100(union — intersection)/union. Ten sets of images
that were correctly matched generated a mean error of 13±2% (mean
± S.D., N=10). In contrast, 10 pairs of randomly matched
images generated a mean error of 19±7% (mean ± S.D.,
N=10). A Kolmogorov—Smirnov test
(Sokal and Rohlf, 1981)
comparing the error distributions in ordered versus randomized
comparisons suggested that they were different (P=0.014). A
qualitative test of the validity of the model was also performed by giving
seven human volunteers 11 MRI shapes and 11 model shapes (taken from sequence
7732, using every other frame from 18 to 38, inclusive) and asking each
volunteer to pair the images so that they matched (a subset of the shapes
presented is shown in Fig.
10). Errors were quantified by computing the difference between
the frame number of a given MRI shape and the frame number of the
corresponding model shape assigned by the human subject and summing this error
over all 11 possible matches. Thus, a subject who assigned all model shapes to
the correct MRI shapes would receive a score of 0, and a subject who
consistently missed all matches by one frame would receive a score of 11.
Actual error scores were 9.6±2.8 (N=7, mean ± S.D.),
suggesting that subjects could match model shapes to MRI shapes within one
frame or better, on average. No subject's match was off by more than two
frames. These results suggest that the model effectively captures changes in
the medio-lateral shape of the buccal mass throughout the feeding cycle.
Buccal mass kinematics
The mid-sagittal and coronal matches between the model and the MRIs suggest
that the overall three-dimensional shape of the buccal mass is captured well
by the model (Fig. 11). The
lateral views of the three-dimensional reconstruction
(Fig. 11A) are in the same
orientation as the original mid-sagittal images
(Fig. 9). The dorsal and
anterior views (Fig. 11B,C)
have been rotated so that the lateral groove (the posterior edge of the model
I3 musculature) is perpendicular to the plane of the page.
Fig. 11A and
Fig. 11B can be compared with
the outputs of the previous odontophore-centric model
(fig. 9G-I and
fig. 9J-L, respectively, in
Drushel et al., 2002). For the
purpose of the discussion of the dimensions of the odontophore, the reference
frame is such that the line defining the attachment of the prow to the I6
muscle is vertical. Thus, the ridge is dorsal, the radular sac is ventral and
the prow is anterior.
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During transition (Fig. 11, left column), the odontophore is widest in the medio-lateral direction and narrowest in the dorso-ventral direction, compared with protraction and retraction, and the radular stalk is deep within the odontophore, suggesting that the radular halves are open. At peak protraction (Fig. 11, middle column), the odontophore is narrower medio-laterally than it is at transition, suggesting that the odontophore may be compressed by the I1/I3/jaw musculature. Shortly thereafter, the radular stalk moves out of the odontophore, suggesting that the radular halves are closing. At peak retraction (Fig. 11, right column), the radular stalk has moved out of the odontophore, which is now longer in the dorso-ventral
shape, i.e. the shape in which the base of the
elongated radula/odontophore extends ventral to the long axis of the buccal
mass (see 
