Vol. 284, Issue 5, H1721-H1728, May 2003
A new method for assessing arteriolar diameter and hemodynamic
resistance using image analysis of vessel lumen
Karel
Tyml1,2,
Donald
Anderson2,
Darcy
Lidington1,2, and
Hanif M.
Ladak2,3,4
1 A. C. Burton Laboratory, Lawson Health
Research Institute, 3 Imaging Research Labs, Robarts
Research Institute, 2 Department of Medical Biophysics,
and 4 Department of Electrical and Computer
Engineering, University of Western Ontario, London, Ontario,
Canada NCA 5C1
 |
ABSTRACT |
To
characterize the nonuniform diameter response in a blood vessel after a
given stimulus (e.g., arteriolar conducted response), frequent serial
diameter measurements along the vessel length are required. We used an
advanced image analysis algorithm (the "discrete dynamic contour")
to develop a quick, reliable method for serial luminal diameter
measurements along the arteriole visualized by intravital video
microscopy. With the use of digitized images of the arteriole and
computer graphics, the method required an operator to mark the image of
the two inner edges of the arteriole at several places along the
arteriolar length. The algorithm then "filled in" these marks to
generate two continuous contours that "hugged" these edges. A
computer routine used these contours to determine luminal diameters
every 20 µm. Based on these diameters and on Poiseuille's law, the
routine also estimated the hemodynamic resistance of the blood vessel.
To demonstrate the usefulness of the method, we examined the character
of spatial decay of KCl-induced conducted constriction along
~500-µm-long arteriolar segments and the KCl-induced increase in
hemodynamic resistance computed for these segments. The decay was only
modestly fitted by a simple exponential, and the computed increase in
resistance (i.e., 5- to 70-fold) was only modestly predicted by
resistance increase based on our mathematical model involving
measurements at two arteriolar sites (Tyml K, Wang X, Lidington D, and
Oullette Y. Am J Physiol Heart Circ Physiol 281:
H1397-H1406, 2001). We conclude that our method provides quick,
reliable serial diameter measurements. Because the change in
hemodynamic resistance could serve as a sensitive index of conducted
response, use of this index in studies of conducted response may lead
to new mechanistic insights on the response.
semiautomatic analysis; conducted response; resistance to blood
flow
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INTRODUCTION |
THE
MEASUREMENT of arteriolar diameter in various tissues
visualized by intravital video microscopy has been the basis of numerous microvascular studies. In intravital studies involving perfusion or superfusion of tissues with pharmacological agents (10, 13), arteriolar responses have been taken to be
fairly uniform along the vessel length, requiring acquisition of
diameter data only at several points along the length to sufficiently
characterize the behavior of the arteriole as a whole. However, in
studies involving a localized application of agents (e.g., via
micropipette), arteriolar responses can vary considerably along the
length (e.g., during conducted arteriolar response; Refs.
5, 15), necessitating a more extensive
diameter analysis to appropriately evaluate the behavior of the arteriole.
Recently, we developed (16) a mathematical model that
predicts behavior of conducting arterioles based on diameter
measurements at two sites (i.e., site of local stimulation and 500 µm
upstream). With the use of Poiseuille's law and the assumption that
conducted response decays exponentially along the arteriolar length
(2), the model characterizes the behavior of arterioles in
terms of the arteriolar resistance to blood flow (i.e., hemodynamic
resistance). However, to our knowledge, the exponential character of
the conducted response has not been verified experimentally, raising
the question of whether the behavior of conducting arteriole can, in
fact, be reliably predicted on the basis of measurements at two sites.
Frequent serial diameter measurements along the vessel length can be
made with the current manual and electronically based techniques
(2, 9, 11, 16). However, the application of these
techniques to frequent serial measurement is tedious and time
consuming. Recently, we (7) developed an image analysis approach to delineate a large blood vessel lumen seen in cross section
(i.e., a closed contour delineation). Because this approach can be
applied to an "open" contour structure represented by the arteriolar wall seen in longitudinal optical section of the arteriole, the major aim of the present study was to use this approach to develop
a quick and reliable method for serial diameter measurements. To
demonstrate the usefulness of the method, we used it to estimate the
hemodynamic resistance of the arteriole and to examine to what degree
this resistance agrees with the resistance predicted by our
mathematical model involving measurements at two sites (16).
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METHODS |
Arteriolar images.
We used images of mouse cremaster muscle arterioles visualized via
intravital video microscopy, as detailed by us previously (16). The procedure for preparation of mouse cremaster was
approved by the Council on Animal Care at the University of Western
Ontario. With a long-working-distance objective (Leitz
×20/0.32 numerical aperture), the images were produced by focusing on
a horizontal plane that included the arteriolar longitudinal axis and
the images of the arteriolar wall whose separation represented the
arteriolar diameter (Fig.
1A). By
focusing up and down, we ensured that the separation between the two
images of the wall indeed represented the arteriolar diameter (i.e., it
was the maximal separation). Microscopic fields of the cremaster muscle
(size 650 × 487 µm) including arteriolar wall images were video
recorded, and selected video frames (Fig. 1A) were digitized
(640 × 480 pixels; 1 pixel = 1.0153 µm) and stored as
computer image files.
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Fig. 1.
Principle of serial diameter measurement.
A: an example of a digitized arteriolar image. Arrow
represents the direction of blood flow. B: an operator uses
computer graphics to place several "dots" along the inner edge of
the "upper" arteriolar wall. The dots represent the initial
vertices of the "upper contour." C: contour-generating
algorithm joins neighboring initial vertices by straight lines to
generate the "initial contour," deletes all but the first and last
vertex, and places on the initial contour dots/vertices spaced equally
(20 µm) apart. Shown are 22 vertices on the initial contour. The
algorithm then undergoes several iterations until the best fit of the
contour (i.e., the position of vertices) to the inner edge is reached
(details in APPENDIX). D and E:
procedure is repeated for the "lower" arteriolar wall.
F: graphic representation of the procedure for diameter
determination. With the "best fit" vertices from both contours,
centroids are determined (details in Fig. 2) and joined by straight
lines (i.e., defined here as the axis segments). At the midpoints of
axis segments, perpendicular lines are drawn to determine diameter
values. G: serial diameter (D) measurements for
arteriole in F. Position 0 is defined as the
midpoint of the first axis segment in the upper right
corner. Bars represent 100 µm.
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Principle of arteriolar contour determination.
In the present study, the arteriolar luminal diameter was defined as
the shortest distance between the inner edges of the arteriolar wall.
We used a computer-based imaging algorithm adapted from recently
published work (7, 8) to represent these edges by two open
contours (details in APPENDIX). In principle, the algorithm for the generation of one contour (i.e., written in Matlab programming language) included four steps. First, the operator used computer graphics to mark the inner edge at several places along the arteriolar length (e.g., the edge in Fig. 1B was marked by 5 dots/vertices). On the basis of this operator's input, the algorithm
then established the "initial" contour (Fig. 1C) with
vertices spaced 20 µm apart (i.e., 20-µm spacing set arbitrarily).
Note that when the total length between the first and the last vertex
was not a multiple of 20 µm, the last two vertices were spaced
by <20 µm. Second, with the use of the digitized image of the
arteriole and an image "forces" equation acting on the initial
vertices (details in APPENDIX), the algorithm underwent
several iterations until a final solution to the forces equation was
reached, yielding new positions of the vertices. The equation included
a gray scale gradient force favoring the greatest light intensity
change at a vertex (i.e., occurring at the luminal edge of the
arteriolar wall) and a "smoothing" force that took into account the
position of the neighboring vertices (i.e., force that counteracted
generation of a jagged contour due to the inherent light intensity
noise present in the image). The third step included a display of the
new vertices allowing the operator to 1) adjust the position
of any of these vertices to best represent the inner edge of the
arteriole (i.e., based on his/her visual judgment) and 2)
run the image forces equation again until a new final solution was
reached. The final fourth step included the operator's "approval"
of the generated contour.
Arteriolar diameter determination.
To evaluate the diameter of a given arteriole, a second contour
representing the other inner edge of the arteriolar wall (i.e., the
"lower" wall in Fig. 1A) was required. Using the same
algorithm, the operator marked the inner edge of the lower wall (Fig.
1D), placing the first and last vertices opposite to the
first and last vertices of the upper contour. The initial lower contour and the new vertices (20 µm apart) were generated (Fig.
1E), a solution to the forces equation was obtained, the
vertices were adjusted, and the contour was approved.
At first glance, serial diameter measurements could be obtained by
measuring the distance between the apposing vertices of the two
contours (Fig. 1F). However, because the first and last vertices of the lower contour were placed only approximately in apposition to the first and last vertices of the upper contour, we
judged this diameter determination to be inaccurate. A better approach
to measure diameter is to determine the longitudinal axis of the
arteriole and then draw a perpendicular "diameter" to it. To
determine this axis, we noted that, within a short segment of the
arteriole, the centroid of vertices of the upper and lower contours
lies on the longitudinal axis of that segment. As shown in Fig.
2, we computed the coordinates of the
centroids from the coordinates of vertices of the two contours. Figure
1F shows the positions of centroids for arteriole of Fig.
1A. We noted that the line segments joining centroids
approximated the longitudinal axis of the arteriole. As shown in Fig.
2, arteriolar diameter at any position along a given line segment of
the axis (e.g., axis segment joining centroids C1 and
C2) could be determined by 1) drawing a line
perpendicular to this segment, 2) determining where this
line intersects the two contours, and 3) measuring the
distance between the two intercepts. In the present study, we chose to
draw this line from the center of the axis segment (Figs. 1F
and 2). Figure 1F shows that, for the arteriole of Fig. 1A, 22 vertices on each contour yielded 21 serial diameter
determinations. (Note that the present diameter routine demanded that
the upper and lower contours have the same number of vertices. For a
substantially curved arteriole, the number of 20-µm-spaced vertices
may not be the same for the two contours. In this case, the contour
with the smaller number of vertices was identified, its first and last vertices were kept, and the remaining vertices were replaced by new
vertices, such that the total number here equaled that of the other
contour. The new vertices still lay on the contour but were spaced by
<20 µm.)
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Fig. 2.
Principle of diameter determination from upper (U) and
lower (L) contours. The positions of vertices of the upper contour
(i.e., U1, U2, U3,...,
Ulast) and of the lower contour (i.e., L1,
L2, L3,..., Llast) are used to
compute the positions of centroids C1, C2,
C3,..., Clast. Thus the x- and
y-coordinates of C1 are computed as the average
of respective x- and y-coordinates of
U1 and L1, whereas the x- and
y-coordinates of C2 are computed as the average
of respective x- and y-coordinates of
U1, U2, U3, L1,
L2, L3. The position of the last centroid is
determined similarly to that of C1, whereas the positions
of all remaining centroids are determined similarly to that of
C2. At midpoints of lines joining centroids, perpendicular
lines are drawn to intercept the upper and lower contours (i.e.,
straight lines joining vertices). A diameter (D) value is
determined as the length of the line between the upper and lower
intercepts.
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Assessment of hemodynamic resistance.
Hemodynamic resistance depends on the blood vessel's geometric and
hemodynamic parameters. It can be estimated from Poiseuille's law,
where it is directly proportional to the viscosity of blood and the
length of the blood vessel but inversely proportional to the vessel
diameter raised to the fourth power. Rather than accurately determining
this resistance based on these parameters, in the present study, we
used Poiseuille's law to conveniently "sum up" the behavior of an
arteriole as a whole, when the diameter along the arteriolar length was
not uniform (e.g., during conducted response).
On the basis of the preceding description of serial arteriolar diameter
measurements, the arteriole of Fig. 1 could be thought of as being
divided into 21 "sleeves." For these sleeves, the hemodynamic
resistance of the ith sleeve, Ri,
could be estimated as
|
(1)
|
where 
is the viscosity of blood (assumed to be 0.03 P),
li is the length of the ith sleeve
(i.e., the length of the longitudinal axis segment between neighboring
centroids; Fig. 1F) expressed in centimeters,
Di is the value (in cm) of diameter drawn
perpendicularly to the ith axis segment (Fig.
1F), and Ri is expressed in poise per
cubic centimeter. The total arteriolar resistance, R, is
then the sum of the 21 sleeve resistances computed from Eq. 1.
Inter- and intraoperator variability.
Because the method depends on the operator's input (Fig. 1,
B and D), we aimed to assess the inter- and
intraoperator variability of the diameter and resistance
determinations. For interoperator variability, six arteriolar images
(500-600 µm in total length) were each analyzed once by three
operators. For intraoperator variability, the same six images were each
analyzed three times by one operator.
Application of method to conducted arteriolar response.
One application of the present method could be its ability to assess
1) the exponential character of conducted response and 2) the degree of agreement between the computed hemodynamic
resistance and the resistance predicted by our model (16).
To this end, we used video recordings from experiments in which mouse
cremaster muscle arterioles were visualized in a control, nonstimulated state and during a conducted response elicited by a 3 M KCl puff ejected from a micropipette for 20-100 ms at 50 psi. During this stimulation, the preparation was superfused by a physiological saline
solution (16) such that the KCl puff could be visualized to be washed away from the upstream portion of the stimulated arteriole. To confirm that the upstream response was not due to diffusion of KCl, we ejected KCl after withdrawing the pipette tip
10-20 µm away from the arteriole. Under these conditions no diameter changes were seen, indicating that conducted response occurred
only after initiation of the local response at the pipette tip. We have
shown (16) that the degree of conducted response in the
mouse cremaster muscle can be affected by a variety of stimuli. Thus,
for the purpose of assessing the exponential character and agreement
between resistances, we analyzed video recordings from four experiments
of good image quality that showed various degrees of conducted response.
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RESULTS |
We used artificial computer-generated arteriolar contours to
verify the proper functioning of our diameter and resistance MatLab
routines (data not shown). Figure 1G shows serial diameter values for the arteriole of Fig. 1A obtained with these
routines. The resistance of this arteriole was computed to be 0.25 × 108 P/cm3 or 25 MP/cm3. With a
standard computer (Hewlett-Packard Vectra VL 400) running at 933 MHz,
it took ~3 min to obtain the two contours, serial diameters, and
resistance. Most of this time was taken by the operator for the
placement of initial vertices and for adjusting the contour to best
represent the inner edge of the arteriolar wall.
Tables 1 and 2 show the results of the
inter- and intraoperator variability
analyses, respectively. For each of the six arterioles, we determined
serial diameters and resistance. On the basis of diameter/resistance
determinations repeated by three operators (or by 1 operator 3 times),
we computed the mean and SE diameter for each of the 25-30 sleeves
along the arteriolar length and the means ± SE resistance for the
whole arteriole. The SE value was taken as a measure of variability.
Tables 1 and 2 show the mean ± SE diameter of the first sleeve,
the SE value averaged among the 25-30 sleeves, and the mean ± SE resistance. In both tables, the SE values of the first sleeve are
comparable to the average longitudinal SE values, indicating that the
variability in diameter measurement of the first sleeve was fairly
representative of the variability of measurement along the arteriolar
length. For the mean diameter of the first sleeve (arterioles
1-6, Tables 1 and 2), the average difference between
individual diameter measurements and this mean was ~1 µm (i.e., for
both inter- and intraoperator repeated measurements). For interoperator
analysis (Table 1), the overall longitudinal SE value among the six
arterioles was 0.63 µm, whereas the overall SE value for the
resistance determination was 4.9 MP/cm3. For intraoperator
analysis (Table 2), the overall longitudinal SE value was 0.46 µm,
whereas the overall SE for resistance determinations was 2.0 MP/cm3. Thus, as expected, there was a tendency for smaller
intra- than interoperator variability of diameter/resistance
determinations (i.e., reflecting better repeatability within operator
rather than between operators in judging the position of inner
arteriolar edge).
A closer inspection of the arteriolar image of Fig. 1A
(i.e., recorded in the control state) reveals a taper over the first 200-300 µm. Figure 1G shows that this change in
diameter amounted to ~10 µm. Serial diameter measurements, averaged
among three operators, in five control arterioles (45- to 60-µm
diameter at midlength) revealed both taper and "peaks and valleys"
along the arteriolar length. This variability in diameter (i.e.,
~5-15 µm between peak and valley) occurred over
~150-450 µm of arteriolar length.
Figure 3A shows an example of
serial diameter measurements (i.e., an average of 3 measurements by 1 operator) in an arteriole analyzed at two time points, immediately
before and 4 s after a local KCl stimulus. Note that both traces
include diameter variability along the arteriolar length, x.
Figure 3B shows the difference between these traces plotted
against x. We fitted this plot with an exponential of the
following form
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(2)
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where 
D0 is the change in diameter at
the local site (i.e., x = 0, the site of maximal
constriction), Dx is the change in diameter
at site x, and 
is the mechanical length constant of the
decay of the conducted response. The resulting fit was 292 µm [i.e., this value was associated with the minimum
root-mean-square (RMS) value of 2.7]. The computed resistances for the
pre- and post-KCl arteriole were Rcontrol = 142 and RKCl = 1,410 MP/cm3,
respectively, indicating an ~10-fold increase in resistance.
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Fig. 3.
An example of serial diameter measurements before and
after KCl stimulus. A: diameter measurements from the same
arteriole immediately before (control) and 4 s after a local KCl
stimulus. Note the spatial variability in diameter at both time points.
B: plot of differences (D) between the serial
diameter measurements shown in A.
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Next, we wanted to examine how this computed resistance increase
compared with the increase predicted by our mathematical model based on
diameter measurements at two sites (16). The model
incorporates the assumption of an exponential decay of the form shown
in Eq. 2. Using the diameter values at x = 0 and x = 542 µm shown in Fig. 3 (i.e.,
D0 = 47 
19 = 28 µm, and
D542 = 48 46 = 2 µm), we
computed predict = 542/ln(2/28) = 205 µm.
Based on this value, the mathematical model (16) predicts the hemodynamic resistance normalized to the control resistance (i.e., that before KCl stimulus), Rnorm,predict,
such that
![R<SUB>norm,predict</SUB><IT>=N</IT><SUP>−1</SUP> <LIM><OP>∑</OP><LL><IT>i</IT>=0</LL><UL><IT>N</IT>−1</UL></LIM>[1<IT>+</IT>(<IT>S−</IT>1)<IT>e</IT><SUP>(−<IT>i/NC</IT>)</SUP>]<SUP>−4</SUP>](/content/vol284/issue5/fulltext/H1721/img003.gif)
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(3)
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where N is the number of sleeves the arteriole is
divided into in this model, S is the diameter at
x = 0 normalized to the diameter before KCl stimulus
(i.e., S = 19/47 = 0.4), and C is predict normalized to the arteriolar length (i.e.,
C = 205/542 = 0.38). Evaluating Eq. 3
with these S and C values and N = 28, we obtained Rnorm,predict = 5.4. Table
3 lists the data for the arteriole of
Fig. 3 (i.e., arteriole 7) and summarizes comparisons between fit and predict and between
"measured" and predicted KCl-induced increases in resistance for
three other arterioles. In general, there was a modest agreement
between fit and predict (i.e., values
were within 3-36% of each other) and between
RKCl/Rcontrol and
Rnorm,predict (i.e., values were within
22-45% of each other). As expected, exponentials incorporating
predict had larger RMS values (i.e., indicating a poorer
"fit" to our D data) than exponentials incorporating
fit.
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Table 3.
Comparison between fitted and predicted spatial decay constants and
between measured and predicted increases in hemodynamic
resistance after local KCl stimulus
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Figures 4 and 5 show diameter and resistance values obtained from
one arteriole. Here, a time series of arteriolar images was obtained by
sampling a video-recorded arteriolar conducted response every ~0.5 s.
Figure 4 depicts a space-time
relationship of this response; note that the pattern of the
longitudinal variability in diameter at time t = 0 (i.e., just before KCl stimulus) appeared throughout the response.
Figure 5A shows this response
at two positions (i.e., x = 0 and x = 600 µm), whereas Fig. 5B shows the time course of the
hemodynamic resistance.
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Fig. 4.
An example of space-time plot of KCl-induced conducted
response. Note that the spatial variability in diameter (i.e., at time
0 s, immediately before the KCl stimulus) appears to persist
throughout the response.
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Fig. 5.
Assessment of conducted response. For the response shown
in Fig. 4, two possible ways of assessment are shown: a conventional
time course of diameter response measured at local (x = 0) and upstream distal (x = 600) sites (A)
and "summed-up" response along the entire arteriolar length using
the computed hemodynamic resistance (B). Note that at the
peak of the response in B, the resistance increased
~40-fold above the baseline.
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DISCUSSION |
The present study adapted an image analysis technique
(7) to generate arteriolar luminal contours from vessel
images recorded via intravital microscopy. On the basis of these
contours, serial diameter measurements were carried out for arterioles
with nonuniform diameters to permit assessment of their hemodynamic
resistance and to capture their behavior as a whole. Although several
methods have been developed to measure the arteriolar diameter at a
given location (1, 9, 11, 12, 14), to our knowledge, the present study describes for the first time a method that can yield quick diameter measurements along the arteriolar length.
Resolution and inter- and intraoperator variability.
To characterize the arteriole as a "whole," we video recorded
arteriolar images at low magnification to analyze as long arteriolar segments as possible. At the present magnification (i.e., field size
650 × 487 µm), the resolution of diameter determination was limited by the pixel size (~1 µm). Thus, when the operator placed contour vertices on apposing inner arteriolar edges (i.e., a procedure equivalent to the manual measurement of luminal diameter), the resolution of the distance measured between these vertices was about
±1 µm. This is the same resolution as that obtained for a manual
measurement reported by us previously (i.e., not involving digitization
of image; Ref. 16). We found that for diameter measurements at a given point along the arteriole, inter- and intraoperator differences between individual measurements and the
corresponding means (Tables 1 and 2) were, on average, about ±1 µm.
Thus the inter- and intraoperator variability in diameter measurement
reflected the present spatial resolution.
In general, the arteriolar image quality (i.e., visibility of the inner
edge of the arteriolar wall) varied along the vessel length. Arteriolar
sites with poorer image quality were associated with higher uncertainty
(both within and between operators) in judging the position of the
inner edge during both initial seeding and final approval of vertices
and also with larger variability of the diameter measurement. Thus,
unfortunately, the effect of image quality at these sites was not
rectified with the contour-generating algorithm. It appears, therefore,
that the present semiautomated method is subject to limitations similar
to those of a manual method (e.g., dependence of visual judgment of the
inner arteriolar edge on image quality) and that the contour-generating
algorithm may not "rescue" the present method from these limitations.
Application of method to conducted arteriolar response.
Although the present method does not eliminate the limitations of
a manual method, it does eliminate the tediousness of repetitive serial measurements. Figure 3 and Table 3 show the outcome of serial
measurement analysis in four arterioles, before and after a KCl
stimulus. As shown in Table 3, there was a modest agreement between
fit and predict values and between
RKCl/Rcontrol and Rnorm,predict values. Discrepancies between
these "measured" and predicted values could be due to the inherent
longitudinal variability in diameter (Figs. 1G,
3A, and 4). This variability was not incorporated in the
computation of predicted values and, for computation of Rnorm,predict in particular, the effect of
variability could be accentuated by the fourth-power relationship of
Poiseuille's law. Although determination of the origin of the
discrepancies/longitudinal variability was not an objective of the
present study, our method may be used in future studies to address this issue.
A typical characterization of a conducted arteriolar response has been
the time course of diameter response at the site of local stimulation
and at some upstream site (Refs. 3, 6; Fig.
5A). Figure 5B shows an alternative
characterization of conducted response in terms of the time course of
hemodynamic resistance of the arteriole. In this example, resistance
has increased ~40-fold above the baseline, indicating that the change
in resistance can be used as a sensitive index of conducted response.
In this context, it is noteworthy that the response of arteriole
7 (Table 3) might be judged as "nonconducting" when using the
typical measure of conducted response of Dx
[i.e., the value of Dx (2 µm) is at the
limit of diameter measurement resolution]. However, on the basis of
serial measurements, this arteriole would be judged as "conducting"
because a substantial 10-fold increase in hemodynamic resistance was
computed. There is evidence that the extent of conducted response
depends on the type of local stimulus used and that multiple cellular
processes may be involved in this response (4). A
sensitive index of the conducted response, such as the change in
hemodynamic resistance, may be required for the analysis of experiments
designed to tease out the different components of this response.
In conclusion, we have used an image analysis algorithm to
develop a method for measuring serial arteriolar diameters and assessing hemodynamic resistance. The method yields quick and reliable
measurements that can be used for analysis of arteriolar responses where diameter changes are not uniform along the length. In
studies of the arteriolar conducted response, the computed hemodynamic
resistance could be used as a sensitive index to characterize the
behavior of the arteriole as a whole.
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APPENDIX |
Here we describe some of the mathematics of the open contour
representation of the inner edge of the arteriolar wall. The representation is based on the work of Lobregt and Viergever
(8), who described a general approach to representing
boundaries in images. The application of their work to the present task
requires the selection of a few algorithmic parameters, and only
relevant mathematical details are given to highlight the required
parameters. Because the contour mathematics undergoes several
iterations before the best "fit" of the contour to the edge is
found, the contour here is referred to as the discrete dynamic contour (DDC).
The DDC consists of vertices that are connected by straight line
segments. When using the DDC, the operator draws an approximate outline
of the desired boundary. The DDC then automatically deforms to
better fit the boundary in the image. The operation of the DDC is based
on simple dynamics. At iteration number t (analogous to
time), a weighted combination of internal
[f 
(t)], image
[f 
(t)],
and damping [f 
(t)]
forces is applied to each vertex i of the DDC, resulting in
a total force
f 
(t)
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(A1)
|
where wimg and
wint are relative weights for the image and
internal forces, respectively. The force causes each vertex to
experience an acceleration. The acceleration is numerically integrated
twice to yield the new position of each vertex. The vertex will
experience different forces at the new location, and Eq. A1
is applied again at the new location. Iterations continue until all
vertices on the DDC come to a rest.
The key to causing vertices to displace toward the inner
arteriolar edge is to design the image forces appropriately. Image forces are defined at vertex i as
![E(x, y)=∥∇[G<SUB>&sfgr;</SUB>∗I(x, y)]∥](/content/vol284/issue5/fulltext/H1721/img009.gif)
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(A2)
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(A3)
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where E represents the "energy" associated with a
pixel having coordinates (x, y),
G
is a Gaussian smoothing kernel with a
characteristic width of 
, and I is the image. The *
operator represents convolution, and 
is the gradient operator. The
symbol || || denotes the magnitude of a vector. The energy
E is the gradient magnitude of the filtered image and has
local maxima at pixels where the gray level changes abruptly. Such an
abrupt change in gray level occurs at the arteriolar luminal edge,
where there is a transition from the dark, blood-filled lumen to the
brightly illuminated arteriolar wall. The force computed from the
energy serves to drive the DDC vertices to the luminal edge. Image
forces have a limited range of effect around an edge. A vertex on the DDC can only be attracted to an edge if it falls in this range. The
spatial extent of this range is determined by in Eq. A2
and becomes larger as becomes larger. Although a large range is desirable to compensate for deficiencies in the initial outlining of
the DDC by the operator (e.g., some vertices on the initial DDC being
outside the range), a large value of can result in poor
localization of the desired edge. A large also offers the advantage
of increased noise suppression. For the images used in this study, we
have selected = 1 pixel (
1 µm) as a compromise.
Referring to Eq. A1, the internal force,
f (t), acts
to minimize local curvature at each vertex, keeping the DDC smooth in
the presence of image noise, whereas the velocity proportional damping
force,
[f (t)], keeps the DDC stable (i.e., prevents oscillations) during DDC deformation;
f (t) and [f (t)] are
defined in the same manner in all applications of the DDC
(8).
In the present study, the weights for the image and internal forces
were chosen to be wimg = 0.9 and
wint = 0.5. Here, the larger weighting for
the image force favors deformation of the contour toward the luminal
edge rather than smoothing due to internal forces.
| |
ACKNOWLEDGEMENTS |
We thank S. Milkovitch, X. Wang, and Drs. C. G. Ellis and I. MacDonald for providing assistance in acquisition of computer image files.
| |
FOOTNOTES |
This work was supported by Canadian Institutes of Health Research
(CIHR) and Heart and Stroke Foundation of Ontario grants awarded to K. Tyml. D. Lidington was a recipient of a CIHR doctoral award.
Address for reprint requests and other correspondence: K. Tyml, Dept. of Medical Biophysics, Univ. of Western Ontario,
London, ON, Canada N6A 5C1 (E-mail:
ktyml{at}lhsc.on.ca).
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.
First published January 9, 2003;10.1152/ajpheart.00741.2002
Received 26 August 2002; accepted in final form 30 December 2002.
| |
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