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ORIGINAL ARTICLE

Determination of vessel cross-sectional area by thresholding inRadon spaceYu-Rong Gao1,2 and Patrick J Drew1,2,3

The cross-sectional area of a blood vessel determines its resistance, and thus is a regulator of local blood flow. However, the cross-sections of penetrating vessels in the cortex can be non-circular, and dilation and constriction can change the shape of the vessels.We show that observed vessel shape changes can introduce large errors in flux calculations when using a single diametermeasurement. Because of these shape changes, typical diameter measurement approaches, such as the full-width at half-maximum(FWHM) that depend on a single diameter axis will generate erroneous results, especially when calculating flux. Here, we present anautomated method—thresholding in Radon space (TiRS)—for determining the cross-sectional area of a convex object, such as apenetrating vessel observed with two-photon laser scanning microscopy (2PLSM). The thresholded image is transformed back toimage space and contiguous pixels are segmented. The TiRS method is analogous to taking the FWHM across multiple axes and ismore robust to noise and shape changes than FWHM and thresholding methods. We demonstrate the superior precision of the TiRSmethod with in vivo 2PLSM measurements of vessel diameter.

Journal of Cerebral Blood Flow & Metabolism advance online publication, 16 April 2014; doi:10.1038/jcbfm.2014.67

Keywords: algorithm; imaging; penetrating vessels

INTRODUCTIONPenetrating arterioles are control points of cortical blood flow1–5

making them important regulators of oxygen delivery to activeneural tissue.6,7 The dynamics of diameter changes in penetratingarterioles can differ from surface vessels,8 consequently accuratemeasurement of the dilatory responses of penetrating vesselsis critical for interpreting functional hemodynamic signals.9 Two-photon laser scanning microscopy (2PLSM) has been employed tomeasure the diameters of penetrating arterioles.4,8 The size of thevessel is typically quantified using the full-width at half-maximum(FWHM) measurement.10,11 An implicit assumption of the FWHMmethod is that the vessels are circular, or nearly circular, althoughsome studies have taken this non-circularity into account whenquantifying vessel sizes.3,12 Careful anatomic quantifications haveshown that blood vessels throughout the body,13 including thecarotid arteries14,15 and coronary veins,16 have decidedly non-circular cross-sections, making a single diameter axis inadequatefor measuring cross-sectional area. Importantly, if very largevessels, which flux a substantial fraction or even all of thecirculating blood, are non-optimally shaped, it is likely that smallervessels will also depart from circularity. In addition, it has beenunderappreciated that the cross-sectional shape of blood vesselscan change with dilation and constriction.17–19 Other factors, suchas mechanical connections to the neighboring tissue20 andchanges in the endothelial cell shape21 can also distort thevessel cross section from circularity and contribute to changes inshape during changes in the vascular tone. Deviations fromcircularity in the vessel lumen cross-section increase vascular

resistance,22 and have widely been postulated to accountfor anomalous measures of viscosity of blood in post capillaryvenules23 and discrepancies in the calculated flow measure-ments through vessels.24–26 While an imaging plane that is notperpendicular to the long axis of the vessel will result in a non-circular cross-section, it cannot account for the changes in theshape observed in these studies.

Rather than assuming circularity and measuring along a singleaxis to determine the cross-sectional area of the vessel, it ispreferable to determine the cross-sectional area of the vesselin a shape-independent manner. One way of measuring areais thresholding27 to generate a binary mask, and quantifyingthe area within the mask. However, thresholding algorithmsare often very sensitive to the threshold value, and perform poorlyon images with low signal-to-noise without more complicatedprefiltering and processing steps.28 As, under physiologic condi-tions, penetrating and ascending vessels have a convexcross-section, we would like to use this constraint to aid our auto-mated image processing algorithms. We make use of the Radontransform,29 in which an image is converted into a series ofprojections at various angles by taking integrals along parallellines. To obtain robust estimates of the area change of the cross-section of vessels oriented perpendicular to the cortical surface,we developed a new method, thresholding in Radon space (TiRS).The TiRS method is analogous to taking the FWHM across manyaxes of an image, such as a frame from movie. Briefly, wetransform the images into Radon space, threshold the image inRadon space, then transform the image back in to image space,

1Center for Neural Engineering, Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, Pennsylvania, USA; 2Neuroscience GraduateProgram, Huck Institutes of the Life Sciences, Pennsylvania State University, University Park, Pennsylvania, USA and 3Department of Neurosurgery, Pennsylvania State University,University Park, Pennsylvania, USA. Correspondence: Professor PJ Drew, Department of Engineering Science & Mechanics, Department of Neurosurgery, W-317 MillenniumScience Complex, Pennsylvania State University, University Park, PA 16802, USA.E-mail: [email protected] work was supported by a Scholar Award from the McKnight Endowment Fund for Neuroscience, a National Scientist Development grant from the AHA, and R01NS078168and R01NS079737 from the NIH to PJD, and ARRA stimulus funds through NS070701.Received 2 January 2014; revised 25 March 2014; accepted 26 March 2014

Journal of Cerebral Blood Flow & Metabolism (2014), 1–8& 2014 ISCBFM All rights reserved 0271-678X/14 $32.00

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where the area of the penetrating vessel is quantified after asecond thresholding, and detection of contiguous pixels.Performing the thresholding in Radon space is advantageousover thresholding in image space, as performing the operation inRadon space makes use of the global structure of the image, notjust a single pixel. Unlike the Hough transform,30 which has beenused to detect parameterized shapes,31 the TiRS method is forquantifying the area of non-parameterized (but convex) objects.We show that the TiRS algorithm is more robust to noise, size, andshape changes than FWHM and standard thresholding methods,and can be applied to real data.

MATERIALS AND METHODSSimulationsAll processing and simulations were done in Matlab (Mathworks, Natick,MA, USA). Within the simulated vessel, the pixel intensity was drawn froman exponential distribution with a mean of one. After varying amounts ofexponentially distributed noise were added, the image was convolved witha 2-dimensional Gaussian with x and y s.d. of one pixel.

For ease of comparison of the FWHM method, which outputs diameteror radius, with the TiRS and pure thresholding methods, which output area,we used the identity as follows: radius¼

ffiffiffiffiffiffiffi

Areap

p . All simulations wererepeated 100 times to determine s.d. In simulations, regions of interest(ROIs) were set to be twice the diameter of the circle used.

Thresholding in Radon SpaceThe Radon transform takes an image and performs line integrals alongparallel lines oriented at an angle y, with the distance of the closest pointalong the line from the center of the image specified by r. r is varied tospan the extent of entire image, and y extends over the range from 0 to180 degrees.

The algorithm for this procedure is given below, and outlined inFigure 3A. The numbers below correspond to images in Figure 3 at variousstages of processing.

(1) We start with a frame of a movie, M[x,y] containing a penetratingvessel. To reduce computational time with in vivo data, and avoidinterference from nearby vessels, a rectangular ROI enclosing the vessel ofinterest is manually selected. The mean pixel intensity, averaged over theentire ROI, is subtracted.

(2) The Radon transform of this image R[r,y] is calculated with y¼ [0,p)in p/180 radian (one degree) increments (Figure 3C, top). The Matlabradon command (Image Processing toolbox) was used for this operation.

(3) For each angle yp, R is rescaled to be between 0 and 1 to givea normalized radon transform Rnorm:

Rnorm½r; yp� ¼R½r; yp� � minðR½r; yp�Þ

maxðR½r; yp� � minðR½r; yp�ÞÞ

(Figure 3C, middle). This rescaling transformation allows us to use asingle threshold for all angles.

(4) For each angle yp, we find the FWHM of Rnorm[r,yp], that is theminimum and maximum r, r� and rþ , where Rnorm[r,yp], respectivelyexceeds and falls below the threshold CRadon. This is analogous to theFWHM operation in image space. Empirically, we found that setting CRadon

between 0.3 and 0.5 effectively segmented the image. We usedCRadon¼ 0.35 for the simulations and measurements shown in this paper.Rnorm[r,yp] is thresholded such that: Rth[r,yp]¼ 1 for r�ororþ , andRth[r,yp]¼ 0 for ror� and rþor (Figure 3B, bottom). The net effect ofthis transformation is to reduce the amplitude of any pixels outside thecentral contiguous shape.

(5) Rth[r,y] is transformed back into image space, using the inverseRadon transform, yielding MIR[x,y] (Figure 3B, right). The Matlab toolboxfunction ‘iradon’ with a Hamming window to reduce high-frequency noisewas used for this transformation.

(6) We threshold Mth[x,y]¼H(MIR[x,y]-c), where H is the Heavisidefunction and c is a constant, typically in the range of 0.1 to 0.3. We used 0.2for simulations and measurements shown in this paper. The area enclosedby the external border of the largest contiguous region of pixels (using8-connectedness, i.e., pixels on the diagonal from each other could becontiguous) is then quantified. This is to ensure that any pixels that arespuriously below this threshold are discounted. A similar procedure is usedin the pure thresholding algorithm (described below).

Pure thresholding algorithmParameters for the pure thresholding algorithm were chosen to optimizeaccuracy under the zero-to-low noise conditions (signal/noiseB3).

(1) We start with a frame of a movie, M[x,y] containing a penetratingvessel. The mean over the entire images is subtracted.

(2) The image is filtered with a 3� 3 pixel median filter. This served toreduce the influence of pixels with very high intensities, increasingthresholding accuracy.

(3) The pixel values in the ROI are normalized to lie between 0 and 1 bythe transformation.

Mnorm½x; y� ¼ M½x; y� �minðM½x; y�ÞmaxðM½x; y� �minðM½x; y�ÞÞ

(4) The pixel intensity is binarized, according to the transformation:

B½x; y� ¼ HnormðM½x; y� � cbtÞ

where H is the Heaviside function and cbt the threshold for binarization.For the simulations shown here, cbt was set to 0.35, with the exception ofFigure 3, where 0.25 and 0.45 were used for the low and high thresholds,respectively.

(5) As with the TiRS method, the area is taken to be largest contiguousregion of pixels (using 8-connectedness).

Full-width at half-maximum(1) The image was first filtered with a 3� 3 median pixel filter. Thisfiltering reduces extreme values, and decreases noise in the diametermeasurement.

(2) A single horizontal axis through the center of the vessel was used todetermine the FWHM. The distance between the most extreme pixelsalong this axis that are 0.5 times the maximum was calculated to be thediameter of the vessel. For ellipse measurements, the diameter chosen wasalong the horizontal axis. This was done purposefully to emphasize themagnitude of error with an arbitrarily chosen axis in a vessel that changesshape.

SurgeryAll surgical and experimental procedures were performed in accordancewith NIH guideline and approved by the Pennsylvania State UniversityInstitutional Animal Care and Use Committee (IACUC). Two-photonimaging subjects were nine (six male) 2- to 4-month-old C57/BL6mice (Jackson Labs, Bar Harbor, ME, USA). Thinned skull windowswere implanted over the right somatosensory cortex after thepolished and reinforced thin-skull windows procedure.32 Briefly, a headbolt was glued to the skull, and three self-tapping #000 3/3200 screws (JIMorris, Southbridge, MA, USA) were threaded into the skull andattached to the head bolt and skull via dental cement. The window areawas first thinned to B30 mm using #7 drill bit, and then polished with 3 fthen 4 f grit (Convington Engineering, Redlands, CA, USA) eachB3 minutes. A #0 cover slip was attached to the polished window areawith cyanoacrylate and the edges sealed with dental cement. Animalswere allowed to recover for 2 days after the surgery before they werehabituated on the imaging setup in 15-minute session for up to foursessions a day.

Two-photon imagingAnimals were imaged using a two-photon microscope consisting of aMovable Objective Microscope (Sutter Instruments, Novato, CA, USA)and a MaiTai HP laser (Spectraphysics, Mountain View, CA, USA), controlledby MPScan software.33 The 20� 0.5 N.A. (Olympus, Center Valley,PA, USA), or 20� 1.0 N.A. (Olympus) water-dipping objectives wereused for imaging. Before each imaging session, animals were brieflyanesthetized with isoflurane (2% in air) and infraorbitally injectedwith 50 mL (50 mg/mL) fluorescein-conjugated dextran (70 kDa; Sigma-Aldrich, St Louis, MO, USA) to visualize vascular system. The mousewas awake and head fixed10 on top of a spherical treadmill equippedwith an optical rotation encoder to record motion.34 Imaging sessionstypically lasted B2 hours. Vessels were imaged for B20 minutes at6 to 9 frames/second. Imaging sessions took place up to 3 months afterthe initial surgery. Vessel cross-sections were imaged 30 to 150 mm belowthe pia.

Diameter measurements of non-circular vesselsY-R Gao and PJ Drew

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Journal of Cerebral Blood Flow & Metabolism (2014), 1 – 8 & 2014 ISCBFM

Data AnalysisAll vessel time series were filtered with a 5-point median filter. All reportednumbers are mean±s.d. unless indicated. All error bars in the plots shows.d. A total of 15 arteries and 11 veins were used in the analysis shownhere. Vessels were chosen from a data set collected for a different set ofexperiments. To allow a fair comparison between the FWHM and TiRSmethods, we excluded vessels that the FWHM method obviously could notresolve the diameter. Post hoc calculation of statistical power (1-b errorprobability) of the flux and diameter discrepancies in Figures 1D and 1Ewas 0.84, and that of the ratios of precision of the FWHM and TiRS methodsin Figure 6F was 0.99.

RESULTSShape Changes in Penetrating Vessels Lead to Large Errors in FluxCalculations using Single Diameter MeasurementFigure 1 shows frames taken from a 2PLSM movie of a singlepenetrating arteriole in the parietal cortex of an awake mouse(Figures 1A and 1B) during a stationary period of no locomotion.This vessel shows obvious changes in cross-sectional area and

shape owing to spontaneous diameter fluctuations.10 Wequantified the non-circularity by taking two measures ofdiameter at perpendicular angles using the FWHM method(Figure 1B, magenta and green lines, respectively). The ratio ofthe two diameters, changes over time (Figure 1C, middle),indicating a shape change in this vessel. We quantified thelong-axis/short-axis ratio, which were placed along the verticaland horizontal axes of the vessels, as in the example (Figure 1B),across a population of penetrating arterioles and ascendingvenules (Figure 1D). The ratios of the measured diameters inthese vessels were substantially different from 1 (Figure 1D). If theimaging plane is not perpendicular to the axis of the vessel, thiswill introduce a minor distortion of the vessel cross-section. Theabsolute area of an off-angle cross-section can be calculated usingtrigonometry. If the angle (y) between the long axis of the vesseland the imaging plane is known, the image can be rescaled bycos(y) along the elongated axis of the image. In practice, thedistortionary effects of off-axis imaging are unlikely to contributesubstantially to the non-circularity seen here, as a 10-degree off-angle will result in a distortion of 2%. We also emphasize that the

Figure 1. Penetrating vessels change shape, making flux calculations based on single diameter measurement highly variable. (A) Schematicshowing location of imaging plane and a vessel. (B) Two frames taken from a movie of a single penetrating arteriole. Images were taken at9Hz nominal frame rate, 50mm below the pia. Both frames are from time points when the mouse was stationary. Time points are indicated bycyan (30 seconds) and purple (35 seconds) arrows in (C), respectively. Magenta (D1) and green (D2) lines indicate the axes along which full-width at half-maximum (FWHM) were calculated. Note the non-circular vessel shape. (C) Time courses of measured diameter and estimatedflow of the penetrating arteriole shown in B by FWHM method. Upper, diameter changes along two axes shown in B. In this vessel, thediameter fluctuates more along the horizontal axis (magenta) than the vertical axis (green). Middle, plot of the ratio of FWHM diameters alongtwo axes (D2/D1), indicating that the vessel changes shape over time. Bottom, flux changes calculated to be proportional to (diameter)4 werehighly dependent on the axis chosen. (D) Two axes diameters ratio calculated by Dlong/Dshort among 15 arteries (red) and 11 veins (blue). Thediameter ratio is plotted versus vessel baseline diameter during quiescence. The ratios of all vessels except one vein are significantly largerthan 1 (one-side t-test, Po0.05 Bonferroni corrected), indicating the significant difference between the estimates obtained along the two axes.The mean of Dlong/Dshort among all the vessels was 1.22±0.27. The median was 1.12. (E) Normalized estimated flux ratio difference of thesame vessels in D. Small difference of diameters measured along different axes can cause huge difference of estimated fluxes. All vessels’normalized estimated flux ratios were significantly larger than 0 (one-side t-test, Po0.05 Bonferroni corrected). The mean of the normalizedestimated flux ratio among all the vessels was 198±371%, median was 58.1%.


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& 2014 ISCBFM Journal of Cerebral Blood Flow & Metabolism (2014), 1 – 8

shape changes cannot be caused by non-perpendicularity of thevessel to the imaging plane, and as motion in the x-y plane isminimal (o1 mm), the changes are unlikely to be due to z-planemotion.

The effects of the non-circularity and shape changes becomeeven more pronounced if we estimate the flux of blood throughthe vessel, which controls local tissue oxygenation.1,2,7–9

According to the Poiseuille equation, the flux of fluid through apipe with radius r is proportional to r4, although this is likely to bea lower bound on the diameter dependence of flow.35,36 Theconsequences of this relationship between radius and flux is thatrelatively small changes in diameter, whether actual or owing tothe measurement noise, will have a dramatic impact on thecalculated flux. To demonstrate how large the effects oncalculated flux might be, we caculated the % difference of themean flux between the two diameter measures across thepopulation of vessels (Figure 1E). The percentage differences inthe flux between the estimates using the diameters measuredalong the two axes were enourmous (Figure 1E), nearly 200%. Thisshows that a single axis of diameter measurement producesunacceptably large variability in both diameter measurments andin blood flux calculations.

The choice of axis can generate enormous variability in thediameter and flux calculations depending on the axis measured.To remove the bias and error intrinsic to the selection of a singlediameter axis, a better method would perform the FWHM along allpossible axes to obtain the average diameter or area. A secondissue is that only a small fraction of the possible pixels is used tocalculate diameter using the FWHM method, and it would bepreferable to use a method that is able to spatially average toreduce noise.

Thresholding in Radon Space Method is more Accurate andPrecise in Measurement of Simulated Penetrating Vessel DiameterTo perform something like the FWHM along multiple diameters ina natural way, we will use the Radon transform.29 An example ofthe Radon transform of a simple binary image of an oval at threesample angles is shown in Figure 2. The advantage of performingthis operation in Radon space is that a FWHM-type calculation caneasily be done at many angles.

In Figure 3, we illustrate the steps involved in the TiRS algorithm(Figure 3A), and show the output along various stages ofprocessing. The numbers in the boxes correspond to the numbersin the corners of the images. The labels on the arrows denote theprocessing step that is done between images. The detailedalgorithm is stated in method. Briefly, the image to be segmentedcontains a circle with a radius of 25 pixels, centered in a 100� 100pixel ROI (Figure 3B left). The raw image is transformed into Radonspace (Figure 3C, top), and the amplitude at each angle is rescaledto lie within 0 and 1 (Figure 3C, middle). The Radon transformedimage is then thresholded (Figure 3C bottom) and transformedback in image space. The transformed image at two differentnoise levels is shown in the right hand column of Figure 3B. Thenet effect of thresholding in Radon space is to boost the contrastof the edges. A second thresholding operation is performed inimage space and the largest contiguous area of pixels isdetermined to be the vessel. Importantly, the detected area isrelatively insensitive to the exact values of these thresholds, unlikethe pure thresholding algorithm, which can be very sensitive tothe exact value chosen (Figure 3D). This demonstrates that underessentially noiseless conditions, the TiRS method easily segmentsthe image, allowing the area of the ‘vessel’ to be easily quantified.

We then compared the effects of adding noise to the image onthe performance of the TiRS algorithm, the FWHM, and the purethresholding algorithms. We used exponentially distributedadditive noise, and quantified the accuracy by comparing theactual area of the circle with that detected by the three algorithms

(Figure 3D, top). At low levels of noise, all algorithms performedwell; however, as the noise increased, the accuracy of both theFWHM and pure thresholding algorithm degraded rapidly. Thefractional error (s.d./true radius) for the TiRS method is lower thanthe other methods when the signal-to-noise ratio is above 1. Thefractional error of the pure thresholding method decreases whenthe noise–to-signal ratio rises above some critical value thatdepends on the threshold parameter. This is because the purethresholding algorithm consistently gives a woefully incorrectestimate of the area of the vessel, as it is incorrectly segmentingnearly the entire image as being above threshold (Figure 3B leftbottom). This can be seen more clearly when we look at the meanabsolute fractional error (Figure 3D bottom). As the noiseincreases, the FWHM method consistently overestimates thediameter of the simulated vessel. The pure threshold method(with a medium threshold parameter) also undergoes a changearound S/N¼ 1, where very large portions of the thresholdedimage were detected as a contiguous region above threshold,resulting in large overestimation of the vessel area. This cansomewhat be mitigated by choosing a higher threshold, but thisresults in inaccuracy at lower S/N ratios (Figure 3D). In practice,where the real diameter is not known, using the pure thresholdingmethod is difficult, as slight changes in the threshold parameterwill give differing measures of diameter, and it will not be clearwhich one is correct. This result clearly shows the robustperformance of the TiRS algorithm to noise, and its superiorityto standard FWHM and pure thresholding operations.

We then address the ability of the TiRS algorithm to accuratelydetect circles of different sizes. This is relevant, as penetratingcortical blood vessels constrict and dilate under natural condi-tions, and our algorithm should be able to detect these cross-sectional area changes. We use a signal-to-noise ratio of 3 forthese simulations, in the regime where pure thresholding andFWHM measures are of comparable accuracy to the TiRSalgorithm. In Figure 4, we plot the performance of these threealgorithms versus the size of the circle to be detected. InFigure 4A, we plot the measured output versus the actual circlediameter for all three methods. All accurately gauge the circle

Figure 2. Radon transform of a convex object permits full-width athalf-maximum, (FWHM) calculations at multiple angles. Upper left,binary image containing a bright oval. Clockwise from the upperright are Radon transforms of this image at 0, 45, and 90 degrees,respectively. Blue lines with arrows show the width of the transformabove the half-maximum threshold (FWHM). Transforms have beenrotated to better show the relationship to the original image.


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diameter. In nearly all conditions, the TiRS method has lower noise(Figure 4B) and is more accurate (Figure 4C) than the FWHM andpure thresholding algorithms. The only exception is for simulatedvessels with very large radii (440 pixels), where the meanfractional error is somewhat higher than the pure thresholdingalgorithm, owing to the constant number of angles used in theRadon for all image sizes. For vessels with diameters of 480 pixelsor so, it is advisable to increase the number of angles to enable amore accurate reconstruction. These issues are only apparent atvessel diameters much larger than those obtained under typicalmagnifications, but bears attention when analyzing real data.

We then address how accurate the TiRS method is indetermining the area when the object to be segmented departs

from circularity. In Figure 5, we parametrically vary the ratio of thelong axis to the short axis of an ellipsoidal region, while keepingthe area constant, to mimic the sorts of shape changes of vesselsseen in vivo. Shape changes in real vessels can be thought of asvariations along the abscissa. Again, we have chosen a signal–to-noise ratio of 3, to give a fair comparison across methods.The FWHM is measured along the horizontal axis to demonstratehow (relatively) slight departures from circularity and shapechanges can cause large errors in area measurements. A non-circular cross-section will cause the FWHM method to give aconstant, potentially very large error in the area, and any changesin shape will cause some variability in this error. In Figure 5A, wesee that both the pure thresholding and TiRS method accurately

Figure 3. Determining penetrating vessel cross-sectional area by thresholding in Radon space is more robust to noise than pure thresholdingor full-width at half-maximum (FWHM). In this and subsequent figures, the results of the thresholding in Radon space (TiRS) method is shownin red, pure thresholding with a low threshold is brown, pure threshold with a medium threshold is orange, pure threshold with a highthreshold is green, FWHM blue, and the true result is a black line. Thick lines show means of 100 runs, shaded areas ± one s.d. (A) Outline ofTiRS algorithm. Numbers in boxes correspond to images at each stage of processing. Labeled arrows denote processing steps. (B) Left, rawimages with two different noise levels, with superimposed detected bounds using the TiRS method (red) or a pure thresholding (mediumthreshold, orange). Right, transformed image after normalizing and thresholding in Radon space. (C) Raw image from top left in (B) at variousstages of processing. Top, Radon transformed image. Middle, image in Radon space after normalization at each angle. Bottom, Radontransforms after thresholding. (D) Effects of noise on the accuracy of the TiRS, threshold and FWHMmethod. Top, measured radius plotted as afunction of noise level for the different methods. Middle, s.d. of measured radius divided by radius for each of the methods. Bottom, meanabsolute fractional error for each of the methods. The FWHM method is the least accurate of all the methods. For a specific thresholdparameter and noise level, the pure thresholding method can perform well, but outside this narrow parameter range, the performancedegrades rapidly. The TiRS method yields the most accurate and precise estimate of vessel size across different noise levels.

Figure 4. Accurate quantification of vessel cross-sectional area for different vessel sizes by the thresholding in Radon space (TiRS) method. Thesignal–to-noise ratio was 3. Thick lines show means of 100 runs, shaded areas ± one s.d. (A) Measured radius versus true radius for the threemethods. (B) S.d. of the measured radius normalized by the true radius, equivalent to the coefficient of variation, showing the TiRS methodhas the lowest variability across all diameters. (C) Mean absolute error versus the radius. At most sizes of vessel, the TiRS method has thelowest absolute error. The full-width at half-maximum method does very poorly on all measures.


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quantify the area of the ellipse across all long/short axes ratios,while the FWHM method is inaccurate and highly variable. Thelow precision and accuracy of the FWHM method is clearly shownin Figures 5B and 5C. The pure thresholding and TiRS methodshave no problem with determining the area of non-circularobjects.

Thresholding in Radon Space Method is more Precise thanFull-Width at Half-Maximum in Quantifying the Diameter and Fluxof Penetrating Vessels Imaged by Two-Photon Laser ScanningMicroscopyFinally, we demonstrate the greater precision of the TiRS methodrelative to the FWHM method on a population of vessels. We didnot compare the pure thresholding algorithm because itssensitivity to the threshold parameter made it difficult to optimizefor use with real data, where signal-to-noise can vary dynamically.Using 2PLSM, we took movies of penetrating vessels in theparietal cortex in an awake mouse head fixed on a sphericaltreadmill34 (Figure 6). This example image shows a simultaneouslyimaged penetrating arteriole and an ascending venule 40 mmbelow the pia (Figure 6B). We used the TiRS method to calculatetheir cross-sectional areas. Frames taken from the movie showchanges in the diameter of both the arteriole and venule, as wellas the contour of area obtained with the TiRS method (Figure 6C,TiRS estimate of arterial in red, venule in blue). Full-width at half-maximum values were taken along the green and magenta axes.The diameters of the two vessels, with TiRS diameter assumed tobe proportional to the square root of area, are plotted inFigure 6D. The FWHM estimates are substantially noisier thanthe TiRS estimate. As with spontaneous and sensory evoked in pialvessels, the artery shows fast dilation and relaxation, on a timescale of seconds, and the vein exhibits smaller dilations that aremuch slower, on the time scale of tens of seconds.10 As the timecourses of the arterial and venous responses are very different, it isunlikely that these changes in cross-sectional area are owing tomovement. Because there is no ‘ground truth’ for the vesseldiameter in real image data, as there was with simulations, wecannot compare the accuracy of the TiRS and FWHM methods forreal data, but we can compare their precision. To quantitativelycompare the precision of the FWHM and TIRS methods, we beganby taking the temporal derivative of the diameters measured withTiRS and the average diameter obtained with two perpendicularFWHM estimates (Figures 6E and 6F). Because the dynamics ofdilation and constriction are slow5,8,10,12 relative to the frame rateused here (6 to 9 Hz), frame-to-frame differences in diameter willprimarily reflect measurement error of the respective methods,not real vessel diameter changes. The temporal derivative of the

diameter obtained with the FWHM method is visibly noisier thanthe TiRS method (Figure 6E). To quantify the relative precision ofthe TiRS and FWHM methods, we plotted the ratios of the s.d. ofthe differentiated FWHM and TiRS traces for the same populationof vessels (Figure 6F) as in Figures 1D and 1E. The ratio can beinterpreted as the relative amount of ‘noise’ in the two methods,with a ratio of 1 implying equal noise, and ratios above 1 meanthat the TiRS method has lower noise. Looking over a populationof vessels, we see a substantial and significant advantage for theTiRS method over the FWHM (Figure 6F). These results demon-strate that the TiRS method gives a much more precise measurethan the FWHM method on in vivo data.

DISCUSSIONThe cross-sectional area of a blood vessel is the prime determinantof its resistance, thus it is important to accurately quantify anychanges in area if we want to understand the regulation of bloodflow.37 Given that the cross-section of blood vessels is non-circular,and that the cross-sectional shape can change, it is critical to use amethod that correctly captures these changes. Here we show thatbecause vessel shape varies and can be non-circular, a standarddiameter method for determining vessel cross-sectional area, thefull-width at half-maximum (FWHM), will give inaccurate andhighly variable results owing to its reliance on a single axis ofmeasurement. Because of this limitation, the FWHM is a poorchoice for quantifying the diameters of penetrating arterioles andascending venules of the cortex. We also showed that athresholding algorithm, when precisely tuned, could performbetter than the FWHM method, but not the TiRS method.However, the accuracy of the thresholding algorithm dependson the signal-to-noise ratio. In practice, the signal-to-noise ratio ofthe imaged vessel could vary dynamically during the course of theimaging session. The signal-to-noise could be changed by dilationof pial vessels,5,10 which can decrease the fluorescence signalowing to the absorption of incoming or emitted photons,5,38

changes in the background owing to out-of-plane fluorescence,39

potentially from dilation of pial vessels containing fluorescent dye,or cross-talk28 from another channel, such as one containing acalcium indicator, whose fluorescence will vary on a rapid timescale. Thus, it is greatly preferable to have an algorithm that givesreliable results in the face of a variable signal-to-noise, such as theTiRS method.

We exploited the global feature sensitivity of the Radontransform to quantify the cross-sectional area of a vessel. TheTiRS method is analogous to taking the FWHM at multipledifferent angles, which gives it two advantages. First, that it canaccurately determine area for convex shapes of varying size, which

Figure 5. Accurate quantification of cross-sectional area during shape changes by the thresholding in Radon space (TiRS) method. The longaxis was varied from 25 to 35 pixels. The short axis started at 25 pixels and was adjusted so that the total area remained constant. The signal-to-noise ratio was 3. Thick lines show means of 100 runs, shaded areas ± one s.d. (A) Ratio of the true area to the actual area for full-width athalf-maximum (FWHM), pure thresholding and TiRS methods. The area in the FWHM case was calculated from the radius, assuming a circularshape, to emphasize how badly the circularity assumption will affect calculated area. Both the pure thresholding and TiRS methods performmuch better than the FWHM method. (B) S.d. of the calculated radius normalized by the true radius versus true radius (calculated from thearea). In addition to being very inaccurate, the FWHM method is highly variable, the TiRS method is the least variable. (C) The TiRS method hasthe lowest mean absolute fractional error, pure thresholding has an intermediate amount, and the FWHM the highest.


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is important as blood vessel cross-sections can be non-circular andcan change their shapes. Second, the TiRS algorithm effectivelymakes use of many measurements of diameter taken at a wide

range of angles, making it very robust to noise, even more so thanpure thresholding algorithms. Image transformations that operateon a global level, such as the Hough and Radon transforms, have

Figure 6. Comparing the results of full-width at half-maximum (FWHM) and thresholding in Radon space (TiRS) methods on in vivo two-photon laser scanning microscopy (2PLSM)-imaged vessel data. (A) An image of the pial vasculature obtained using 2PLSM. The red and bluearrows indicate the imaged penetrating arteriole and ascending venule, respectively. Both vessels were imaged 40 mm below the pia. (B) X–Zmaximum projections of the penetrating arteriole and ascending venule in A. (C) Frames taken from the movie during periods of quiescence(125 seconds) and locomotion (238 seconds). Images were taken at a rate of 8.3 frames per second. Green and magenta lines indicate the twoaxes where the diameters were calculated by FWHM. Red and blue contours show the vessel bounds detected using the TiRS method.(D) Time courses of diameter changes of the penetrating arteriole (upper panel) and ascending venule (bottom panel) obtained from twosingle axes (magenta and green traces) and using the TiRS method (artery-red, vein-blue). Black dots denote locomotion. Note the shapechange of both vessels during locomotion. (E) Temporal derivatives of diameter time series in D. Gray traces are the temporal derivatives ofaveraged FWHM diameter measures along vertical and horizontal axes of the artery (upper panel) and vein (bottom panel). Red and bluetraces correspond to the TiRS measurement of arteriole and venous diameters, respectively. Note the larger noise from the FWHM methodthan with the TiRS method. (F) Histogram of the s.d. ratios of the temporal derivatives of two measurements. Gray vertical line indicates a ratioof 1. The mean of the s.d. ratios among all vessels was 3.08±2.70, and median was 2.11. The ratios of all vessels were significantly larger than 1(one-side t-test, Po0.05 Bonferroni corrected).


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been extensively used in computer vision to detect and localizeparticular parameterized shapes,31 or to extract a parameter fromthese shapes, such as the slope of a line.40 By using the globalnature of the Radon transform to suppress noise, we can obtainmore accurate measures of area of an object than other methods.The TiRS method’s robustness to noise will likely prove helpful instudying vascular dynamics that cannot be averaged over trials,such as spontaneous dilations.10

DISCLOSURE/CONFLICT OF INTERESTThe authors declare no conflict of interest.

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