## 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

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).

## 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.1
*A*). 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. 1
*A*) were digitized (640 × 480 pixels; 1 pixel = 1.0153 μm) and stored as computer image files.

#### 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
). 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. 1
*B* was marked by 5 dots/vertices). On the basis of this operator's input, the algorithm then established the “initial” contour (Fig. 1
*C*) 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
), 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. 1
*A*) was required. Using the same algorithm, the operator marked the inner edge of the lower wall (Fig.1
*D*), 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.1
*E*), 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. 1
*F*). 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. Figure1
*F* shows the positions of centroids for arteriole of Fig.1
*A*. 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 C_{1} and C_{2}) 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. 1
*F*and 2). Figure 1
*F* shows that, for the arteriole of Fig.1
*A*, 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.)

#### 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 *i*th sleeve, *R _{i}
*, could be estimated as
Equation 1where η is the viscosity of blood (assumed to be 0.03 P),

*l*is the length of the

_{i}*i*th sleeve (i.e., the length of the longitudinal axis segment between neighboring centroids; Fig. 1

*F*) expressed in centimeters,

*D*is the value (in cm) of diameter drawn perpendicularly to the

_{i}*i*th axis segment (Fig.1

*F*), and

*R*is expressed in poise per cubic centimeter. The total arteriolar resistance,

_{i}*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.

## RESULTS

We used artificial computer-generated arteriolar contours to verify the proper functioning of our diameter and resistance MatLab routines (data not shown). Figure 1
*G* shows serial diameter values for the arteriole of Fig. 1
*A* obtained with these routines. The resistance of this arteriole was computed to be 0.25 × 10^{8} P/cm^{3} or 25 MP/cm^{3}. 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/cm^{3}. 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/cm^{3}. 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. 1
*A*(i.e., recorded in the control state) reveals a taper over the first 200–300 μm. Figure 1
*G* 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 3
*A* 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 3
*B* shows the difference between these traces plotted against *x*. We fitted this plot with an exponential of the following form
Equation 2where Δ*D*
_{0} is the change in diameter at the local site (i.e., *x* = 0, the site of maximal constriction), Δ*D _{x}
* is the change in diameter at site

*x*, and λ is the mechanical length constant of the decay of the conducted response. The resulting λ

_{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

*R*

_{control}= 142 and

*R*

_{KCl}= 1,410 MP/cm

^{3}, respectively, indicating an ∼10-fold increase in resistance.

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., Δ*D*
_{0} = 47 − 19 = 28 μm, and Δ*D*
_{542} = 48 − 46 = 2 μm), we computed λ_{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), *R*
_{norm,predict}, such that
Equation 3where *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 λ_{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 *R*
_{norm,predict} = 5.4. Table3 lists the data for the arteriole of Fig. 3 (i.e., *arteriole 7*) and summarizes comparisons between λ_{fit} and λ_{predict} and between “measured” and predicted KCl-induced increases in resistance for three other arterioles. In general, there was a modest agreement between λ_{fit} and λ_{predict} (i.e., values were within 3–36% of each other) and between*R*
_{KCl}/*R*
_{control} and*R*
_{norm,predict} (i.e., values were within 22–45% of each other). As expected, exponentials incorporating λ_{predict} had larger RMS values (i.e., indicating a poorer “fit” to our Δ*D* data) than exponentials incorporating λ_{fit}.

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 5
*A* shows this response at two positions (i.e., *x* = 0 and *x* = 600 μm), whereas Fig. 5
*B* shows the time course of the hemodynamic resistance.

## 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 λ_{fit} and λ_{predict} values and between*R*
_{KCl}/*R*
_{control} and*R*
_{norm,predict} values. Discrepancies between these “measured” and predicted values could be due to the inherent longitudinal variability in diameter (Figs. 1
*G*,3
*A*, and 4). This variability was not incorporated in the computation of predicted values and, for computation of*R*
_{norm,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.5
*A*). Figure 5
*B* 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 Δ*D _{x}
*[i.e., the value of Δ

*D*(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.

_{x}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.

## Acknowledgments

We thank S. Milkovitch, X. Wang, and Drs. C. G. Ellis and I. MacDonald for providing assistance in acquisition of computer image files.

## 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*)
Equation A1where *w*
^{img} and*w*
^{int} 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. EA1
*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
Equation A2
Equation A3where *E* represents the “energy” associated with a pixel having coordinates (*x*, *y*),*G*
_{ς} is a Gaussian smoothing kernel with a characteristic width of ς, 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 ς in *Eq. EA2
*and becomes larger as ς 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 ς can result in poor localization of the desired edge. A large ς also offers the advantage of increased noise suppression. For the images used in this study, we have selected ς = 1 pixel (≈1 μm) as a compromise.

Referring to *Eq. EA1
*, 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 *w*
^{img} = 0.9 and*w*
^{int} = 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.

## 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

- Copyright © 2003 the American Physiological Society