Measurement and classiﬁcation of retinal vascular tortuositysumbaug/RetinalProjectPapers/Measurement … · Keywords:Tortuosity; Retina; Blood vessel; Automated measurement 1. Introduction

International Journal of Medical Informatics 53 (1999) 239–252

Measurement and classification of retinal vascular tortuosity

William E. Hart a,*, Michael Goldbaum b, Brad Cote c, Paul Kube d,Mark R. Nelson e

a Department of Applied and Numerical Mathematics, Sandia National Laboratories, Alburquerque, NM 87185, USAb Department of Ophthalmology, Uni6ersity of California, San Diego, CA, USA

c The Registry, Inc., San Diego, CA, USAd Department of Computer Science and Engineering, Uni6ersity of California, San Diego, CA, USA

e Data Vector, San Diego, CA, USA

Abstract

Automatic measurement of blood vessel tortuosity is a useful capability for automatic ophthalmological diagnostictools. We describe a suite of automated tortuosity measures for blood vessel segments extracted from RGB retinalimages. The tortuosity measures were evaluated in two classification tasks: (1) classifying the tortuosity of bloodvessel segments and (2) classifying the tortuosity of blood vessel networks. These tortuosity measures were able toachieve a classification rate of 91% for the first problem and 95% on the second problem, which confirms that theycapture much of the ophthalmologists’ notion of tortuosity. Finally, we discuss how the accuracy of these measurescan be influence by the method used to extract the blood vessel segments. © 1999 Elsevier Science Ireland Ltd. Allrights reserved.

Keywords: Tortuosity; Retina; Blood vessel; Automated measurement

1. Introduction

Normal retinal blood vessels are straight orgently curved. In some diseases, the bloodvessels become tortuous, i.e. they become di-lated and take on a serpentine path. Thedilation is caused by radial stretching of theblood vessel and the serpentine path occurs

because of longitudinal stretching. The tortu-osity may be focal, occurring only in a smallregion of retinal blood vessels, or it mayinvolve the entire retinal vascular tree. Fig. 1shows images with tortuous and non-tortu-ous blood vessels.

Many disease classes produce tortuosity,including high blood flow, angiogenesis andblood vessel congestion. High blood flowmay occur locally in arteriovenous anastamo-* Corresponding author. E-mail: [email protected].

1386-5056/99/$ - see front matter © 1999 Elsevier Science Ireland Ltd. All rights reserved.

PII: S 1 3 8 6 - 5 0 5 6 ( 9 8 ) 0 0 1 6 3 - 4

W.E. Hart et al. / International Journal of Medical Informatics 53 (1999) 239–252240

Fig. 1. Images with (a) tortuous and (b) non-tortuous blood vessel segments.

sis or systematically in high cardiac outputsuch as accompanies anemia. In response toischemia or inflammation, new blood vesselsgrow, often in a destructive manner. Theangiogenic process commonly causes dilationand tortuosity of retinal blood vessels priorto the onset of the neovascularization.Venous congestion can result from the occlu-sion of the central retinal vein or a branchretinal vein. The resulting elevated intravas-cular pressure results in tortuosity and dila-tion in the blocked vein.

Information about disease severity orchange of disease with time may be inferredby measuring the tortuosity of the bloodvessel network. Consequently, there is abenefit in measuring tortuosity in a consis-tent, repeatable fashion. Vascular tortuositymeasurements are commonly performed byhuman observers using a qualitative scale(e.g. mild, moderate, severe and extreme).Variability in grading can result because theboundaries between the grades may differ

between observers and with the same ob-server at different times. Furthermore, mea-surements on such a gross scale makechanges in vascular tortuosity difficult todiscern.

A quantitative tortuosity measurement wasfirst described by Lotmar, Freiburghaus andBracher [1] and slightly extended by Bracher[2]. This measurement involves the manualselection of points on fundus photographsthat subdivide the tortuous vessel into a se-ries of single arcs. Fig. 2 illustrates the mea-

Fig. 2. Illustration of the subdivision used to estimateblood vessel tortuosity with the relative length varia-tion.

W.E. Hart et al. / International Journal of Medical Informatics 53 (1999) 239–252 241

surements they use to calculate the vessel’stortuosity. Their method decomposes the ves-sel into a series of circular arcs, for which thechord lengths li and arrow heights hi aremeasured. The tortuosity is measured as therelati6e length 6ariation

L− ll:

83

%i

�hi

li

�2

,

where L is the length of the blood vessel andl is the chord length. The approximation isderived using a sinusoidal model of a bloodvessel. This method was used by Kylstra etal. [3] to measure the effects of transmuralpressure on the tortuosity of latex tubing.

Given the increased availability of digitizedfundus photographs, automated tortuositymeasurements are now feasible. Kaupp et al.[4] have reported unpublished results of anautomated tortuosity measurement that usesa Fourier analysis of the perpendicular alongthe blood vessel. Smedby et al. [5] describefive tortuosity measures used to measure tor-tuosity in femoral arteries. Included are sev-eral measures of the integral curvature alongthe blood vessel, the number of inflectionpoints of the vessel and the fraction of thevessel that has high curvature. Their experi-ments examine properties of these measureslike reproducibility and scalability. Ca-powski, Kylstra and Freedmen [6] describe ameasure of blood vessel tortuosity based onspatial frequencies. Zhou et al. [7] have alsodescribed a method for distinguishing tortu-ous and nontortuous blood vessels in an-giograms. In a preliminary abstract, we haveproposed a tortuosity measure based on theintegral curvature along a blood vessel [8].

In this paper we describe tortuosity mea-sures that are used to measure the tortuosityof retinal blood vessels as well as the retinalblood vessel network. We motivate our tortu-osity measures by analyzing abstract proper-ties of measures based on medical intuitions

of tortuosity. To assess the relative utility ofthese measures, they were used to classifyblood vessel segments and blood vessel net-works. The segments used in these classifica-tion experiments were extracted manuallyand automatically and we discuss how thetortuosity measures can be influenced byproperties of the method used to extract thevessel segments. The classification rate was ashigh as 91.5% for blood vessel segments and95% for blood vessel networks. While nosingle tortuosity measure was clearly supe-rior, the experiments recommend a totalsquared curvature measure.

2. Tortuosity measures

2.1. Abstract properties

We begin by discussing several propertiesof tortuosity measures that are motivated bythe ophthalmologist’s notion of tortuosity. Atortuosity measure that has many of theproperties described below should providegood predictive performance in our classifica-tion tasks, thereby matching an ophthalmolo-gist’s measure of tortuosity. We will discussthe following properties of tortuosity mea-sures: invariance to translation and rotation,response to scaling and compositionality ofvessels and vessel networks. These propertiesare defined for parametrized differentiablecurves C= (x(t), y(t)), with t in an interval[t0, t1]. We model retinal blood vessels withsuch curves. A tortuosity measure t takes acurve C as its argument and returns a realnumber.

2.1.1. In6ariance to translation and rotationSuppose a vessel with tortuosity t were

moved on the retina without changing itsshape or size. What should be the tortuosityof the resulting vessel?


While ophthalmologists’ judgments of ves-sel tortuosity do seem to incorporate therelative curvature of other vessels on the fun-dus, we believe their judgment is largely inde-pendent of the location or orientation of thevessels. Our measures assume that vessel tor-tuosity is not dependent on the location oforientation of the vessel.

2.1.2. Response to scalingSuppose a vessel with tortuosity t was

enlarged uniformly by some scale factor g\1. What should be the tortuosity of the scaledvessel?

Ophthalmologists do not seem to have un-equivocal intuitions about this question,though they believe that a vessel’s tortuosityshould be roughly invariant to scaling. How-ever, it is clear that if scale does affect thetortuosity then it does so in a multiplicativemanner. That is, if curve C= (x(t), y(t)) hastortuosity t(C), curve C %= (gx(t), gy(t)) hastortuosity b(g) · t(C) for some function b.If tortuosity is invariant to scaling, b(g) 1;and if tortuosity is inversely related to scale,b(g)B1 when g\1. We examine tortuositymeasures that exhibit various tortuosity scalefactors b(g) and investigate their effect onclassification performance.

2.1.3. Vessel compositionalitySuppose a vessel were composed of two

vessel segments C1 and C2, one with tortuos-ity t(C1) and the other with tortuosity t(C2).What should be the tortuosity of the wholevessel?

It appears to be the intuition of ophthal-mologists that a vessel that is composed oftwo segments of different tortuosities wouldhave a degree of tortuosity between the tortu-osities of its constituent segments. This as-sumes that the two segments are smoothlyconnected, but the ‘order’ in which segmentsoccur in the vessel is not important. It also

appears that if a vessel segment with a giventortuosity were extended with a segment ofthe same tortuosity, the resulting vesselwould have the same tortuosity as either ofits segments. That is, a vessel doesn’t increase(or decrease) its tortuosity by having thesame serpentine pattern extended at greaterlength.

We formalize these intuitions as follows.We write C=C1�C2 and say curve C hasconstituent segments C1 and C2. WheneverC= (x(t), y(t)) with t� [t0, t1], C1= (x(t),y(t)) with t� [t0, t2] and C2= (x(t), y(t)) witht� [t2, t1], for some t2� [t0, t1]. Now supposethat t(C1)5t(C2). We say that a tortuositymeasure has the property of compositionalitywhen, for any such C1, C2, C3,

t(C1)5t(C)5t(C2) (1)

with the equality holding if and only ift(C1)=t(C2).

The property of compositionality providesconstraints on how to compute the tortuosityof a vessel, given the tortuosities of its con-stituent segments. This computation will benecessary in an automated system that ex-tracts vessel segments and not whole vesselsfrom fundus images. A method of computingthe tortuosity of a whole vessel that is consis-tent with Eq. (1) is to weight the tortuosity ofeach constituent segment by the fraction ofarc length which that segment contributes tothe vessel. That is, t(C1�C2)= [s(C1) t(C1)+s(C2) t(C2)]/s(C1�C2), where s(C) is thearc length of the curve C (see below). We callthis method weighted additivity.

On the other hand, computing the tortuos-ity of a vessel by averaging the tortuosities ofits constituent segments is inconsistent withEq. (1). It follows from Eq. (1), together withthe assumption that the tortuosity of a vesselis independent of how it is segmented, thatthe tortuosity of a vessel cannot be a functiononly of the tortuosities of its constituent seg-


Fig. 3. Two curves with the same tortuosity for ameasure with chord-colinear compositionality.

colinear and whose tortuosity is roughlyequivalent.

2.1.4. Network compositionalityAnother type of compositionality concerns

the way in which the tortuosity measures forvessel segments are combined to determine atortuosity measure for an entire vessel net-work. This type of compositionality mightdiffer from vessel compositionality becausethe method for combining the tortuosity val-ues from vessel segments may depend onwhether they are part of the same vessel. Wehave yet to develop a method for extracting acomplete vessel network, so we calculate thetortuosity of a blood vessel network using theweighted additivity of all of the blood vesselsegments in the image.

2.2. Definitions

Using our previous definition of a curvesegment C, we define a suite of tortuositymeasures. We begin by defining the compo-nents used to construct these measures. Thesemeasures are defined for a curve C= (x(t),y(t)) on the interval [t0, t1].

2.2.1. Arc lengthThe arc length of C is

s(C)=&

t 0

t1

x %(t)2+y %(t)2 dt.

2.2.2. Chord lengthThe chord length of C is

chord(C)=(x(t1)−x(t0))2+ (y(t0)−y(t1))2

2.2.3. Total cur6atureThe curvature of C at t is

k(t)=x %(t)y¦(t)−x¦(t)y %(t)

[y %(t)2+x %(t)2]3/2 , (2)

and the total curvature of a curve segment Cis

ments. Let t(C1)=t(C2)Bt(C3)=t(C4) andconsider the curves C1�C2�C3 and C2�C3�C4. If we segment these curves asC1� (C2�C3) and (C2�C3)�C4, then thetortuosity measures for these two curves aredifferent. However, if these curves are seg-mented as (C1�C2)�C3 and C2� (C3�C4),the tortuosities of the constituent segments ofthe two vessels are the same, viz. t(C1) andt(C3).

Finally, we note that the property of com-positionality implies that extending a vesselsegment ‘in the same way’ does not affect itstortuosity. The vessel segment which consti-tutes the extension would have the same tor-tuosity as the vessel segment, so thetortuosity of the extended vessel remains thesame. Some of the proposed tortuosity mea-sures will be invariant to some types of exten-sions without strictly satisfying thecompositionality property. We say that ameasure has the property of chord-colinearcompositionality if a vessel C is segmentedsuch that each segment has the same tortuos-ity and the chords of the segments are colin-ear, then the tortuosity of the vessel is thesame as the constituent segments. A tortuos-ity measure with this property is invariant tochord-colinear extensions. Fig. 3 shows twocurves that will have the same tortuosity for ameasure that has chord-colinear composition-ality. This is an interesting property sinceretinal blood vessels are often roughly peri-odic along a straight line. Thus they can bepartitioned into segments whose chords are


Table 1Summary of the tortuosity measures and their properties

CompositionalityChord-colinear compositionalityTortuosity measure Response to scale

1 ãt1 s(C)/chord(C)−11/gt2 tc(C)1/g2t3 tsc(C)1/g2 ã ãt4 tc(C)/s(C)

tsc(C)/s(C) 1/g3 ãt5 ã1/g2tc(C)/chord(C) ãt6

1/g3 ãt7 tsc(C)/chord(C)

tc(C)=&

t 0

t1

�k(t)� dt.

2.2.4. Total squared cur6atureThe total squared curvature of C is

tsc(C)=&

t 0

t1

k(t)2 dt.

Table 1 defines the measures that we exam-ined in our experiments and describes theirproperties. These measures have zero mea-sure for straight vessel segments and increas-ing positive measure for segments as theybecome tortuous. The variety of tortuositymeasures allows us to compare measures todetermine the relative importance of the tor-tuosity properties described previously.

The measure t1 simply computes the tortu-osity of the segment by examining how longthe curve is relative to its chord length. Thismeasure is the same as the distance factortortuosity measure described by Smedby etal. [5] and it is very similar to the relativelength variation proposed by Lotmar et al.[1]. If the curve C is a single arc, or if C ischord-colinear, then l=chord(C) and t1 isequivalent to relative length variation.

Measures t2 and t3 directly calculate thecurvature of the curve. Unfortunately, nei-ther of these measures have either of thecompositionality properties. The t3 measurediffers from t2 in that it places a greater

emphasis on the parts of the curve that havehigh curvature and de-emphasizes the partsof the curve that have low curvature. Sincethe curvature is greater for small vessels, t3

will emphasize the tortuosity of smaller ves-sels more than t2. The t2 measure is the sameas the total curvature measure described bySmedby et al. [5].1

The remaining measures are ‘length-nor-malized’. Measures t4 and t5 average the totalcurvature measures by the arclength, while t6

and t7 average by the chord length. Themeasures t4 and t5 have the advantage thatthey have the property of compositionality.We believe that these two measures a prioricome closest to the ophthalmologist’s notionof tortuosity.

2.3. Tortuosity calculation

The definitions for our tortuosity measuresapply to abstract curves. The skeletonizedblood vessels in our data sets consist of se-quences of pixel locations that approximatethe center line of the blood vessels in theretinal images. The arc length of a skele-tonized blood vessel with n pixel locations

1 We disagree with the claim by Smedby et al. that the totalcurvature measure is invariant to linear scaling. If x(t) andy(t) are multiplied by a constant factor, A, then their deriva-tives are multiplied by A. Thus the curvature measure in Eq.(2) is scaled by 1/A.


(xi, yi) can be calculated by walking along thecurve and summing the distance betweenneighboring points:

%n−1

i=1(xi−xi+1)2+ (yi−yi+1)2.

To calculate the total curvature and totalsquared curvature, recall [9] that the curva-ture of a parameterized curve can also bedefined as

k(t)=da

ds(t),

where a(t) is the angle of the tangent line att. To compute (da/ds)(t), we measured a(t)and s(t) and computed the derivative of a(t)with respect to s(t). Here, s(t) is the arclength between (x1, y1) and (xt, yt) We deter-mined a(t) by calculating the angle of the lineconnecting (xt, yt) and (xt+ l, yt+1). To com-pute the derivative (da/ds)(t), we used a win-dow of 10 values of s(t) and a(t): (s(t−4),a(t−4)),…, (s(t+5), a(t+5)). A line,f(x)=mx+b, was fitted to these points witha linear regression method to minimize

%i

(a(i)− f(s(i)))2.

The derivative (da/ds) was equated with m,the slope of this line. The n−9 values for(da/ds) were used to numerically estimate theintegrals needed to calculate the total curva-ture and total squared curvature.

Following a suggestion of Flynn and Jain[10], we smoothed the pixel representation ofa blood vessel segment before making ourtortuosity calculations. We independentlysmoothed the x and y coordinate sequences,{x1,…, xn} and {y1,…, yn} using the smooftmethod described by Press et al. [11], whichapplies a low-pass filter to the data. Thismethod eliminated undesirable noise that isdue to the discrete nature of the pixel repre-sentation. For example, a blood vessel at a

45° angle in a image can have a pixel repre-sentation that is a zigzag line of pixels alongthe blood vessel. The smoothed coordinatesare not restricted to the discrete pixel grid, sotheir tortuosity measures more closely reflectthe tortuosity of the actual blood vessel. Pre-liminary experiments indicated that two ap-plications of this smoothing method weresufficient to reduce this noise.

The smoothing operation also eliminatednoise that was introduced when independentsegments were linked together to form alonger segment. Adjacent segments extractedfrom a blood vessel can form a non-smoothsequence of pixels. Smoothing combined seg-ments produced a sequence of points thatmore closely resembled the contour of theoriginal blood vessel.

3. Methods

We examined the utility of our tortuositymeasures by using them as features in twodifferent classification problems. In the firstproblem, we classified blood vessel segmentsas tortuous or non-tortuous. In the secondproblem, we classified the tortuosity of theentire blood vessel network.

3.1. Data

Blood vessel segments were extracted froma set of 20 retinal images in two differentways: automatically and manually. To extractblood vessel segments, we applied the bloodvessel filter described in Chaudhuri et al. [12]to the green plane of an RGB image; thegreen plane was selected because it typicallyexhibits the greatest contrast. The filter wasapplied at 12 orientations over 180°. The finalresponse map was computed by taking themaximum response of the 12 filters at eachlocation. We thresholded and thinned the


response map of the blood vessel filter toproduce an image containing binary edge seg-ments. The edge segments were primarilyblood vessels, but also included some edgesof large objects like the optic nerve.

To create the set of automatically extractedblood vessel segments, the edge segmentswere classified as blood vessels or non-bloodvessels using the linear classifier described inCote et al. [13]. There were 981 automaticallyextracted blood vessel segments, of which 252were tortuous and 729 were non-tortuous.

The unclassified edge segments were alsoused to extract blood vessel segments manu-ally. The edge segments were manually iden-tified as blood vessels and linked together toform the final blood vessel segments. Carewas taken to link vessel segments only if thesmoothness of the link reflected the curvatureof the underlying blood vessel. This is reason-able since blood vessel segments aresmoothed before the tortuosity measures arecalculated. There were 284 blood vessels ex-tracted manually, of which 133 were tortuousand 151 were non-tortuous.

We are primarily interested in the automat-ically extracted blood vessels, since they canbe used as part of automated diagnostictools. However, the blood vessel filter tendsto break up longer blood vessels. This usuallyoccurs when two vessels cross in an image,where a vessel branches or when a vesselbends very rapidly. Since tortuous blood ves-sels can bend very rapidly, they are brokenup more often than non-tortuous vessels. Themanually extracted blood vessel segmentsprovide a benchmark against which the per-formance of the automatically extracted seg-ments can be compared.

After extraction, the tortuosity measureswere calculated for each segment and thesegments were labeled by one of the authors(MG), a retinal specialist who has experiencewith diseases causing tortuosity in retinal

blood vessels. These two sets of extractedblood vessels were used for the first classifica-tion problem.

For the second classification problem, theextracted blood vessels were used to calculatethe network tortuosity measures for each im-age. The vessel network tortuosity of the 20retinal images were labeled; there were tentortuous and ten non-tortuous images.

3.2. Classification

3.2.1. Logit modelsWe used a logit model [14] to classify the

data for both classification problems. Forproblems with two classes, a logit modelcomputes a weighted sum of the input fea-tures passed into a logistic function. Let{ f1,…, fn} be n input features and {w0,…,wn} be the n+1 weights. The two-class logitmodel is

f(x, w)=g�

w0+ %n

i=1wi fi

�,

where g(x) is the logistic function g(x)=1/(1+e−x). The output of the logit model isbetween zero and one. To perform classifica-tion, the output is thresholded to zero or one,depending on whether the output is greateror less than 0.5.

Given a set of training examples,{(x1, y1),…, (xn, yn)}, the optimal weight vec-tor is found by minimizing

J(w)=%i

E(yi h, f(xi, w))

where

E(a, b)=− log(1−b) , a=0

− log(b) , a=1

This is the ‘cross-entropy’ error, which ismotivated by an analysis based on the rela-tive entropy of the distribution of outputs

ÍÁ

Ä


with the distribution of binary data [15]. Wefound the optimal weight vector by minimiz-ing J(w) with the conjugate gradient method[11].

Logit models are well suited for modelingthe probability distribution of binary data. Ifthe data are linearly separable, logit modelswill simply perform a linear classification ofthe data. However, for data that are notlinearly separable, logit models perform dif-ferently from linear models. Aldrich and Nel-son [16] describe how nonlinear transforma-tions like the logistic function modify thebehavior of the linear model. Further, theynote that the there are a variety of reasonswhy the assumption that a probability modelis linear is unrealistic in most cases. In prelim-inary experiments, we found that logit modelshad better classification performance than lin-ear models.

3.2.2. Performance measuresTwo measures were used to compare the

performance of classifiers using the differenttortuosity measures: classification rate andthe integrated relative operating characteristic(ROC). The classification rate is simply theproportion of test samples that are correctlyclassified. The ROC measures the proportionof test samples that are correctly classified aspositive instances (true-positives) as a func-tion of the proportion of negative test samplesthat are classified as positive instances (false-positives) [17]. The integrated ROC measureshow a classifier’s performance varies for dif-ferent classification thresholds. The integratedROC, A, typically assumes values between 0.5and 1.0. When A is near 0.5, the classifierperforms no better than chance alone. WhenA is near 1.0, the classifier has near perfectdiscrimination for most classificationthresholds.

The ROC was measured by calculating theoutput of the logit model for all of the test

samples. The outputs were split into 20groups based on the rank of their values. Thetrue-positive and false positive proportionswere measured for each group. The trape-zoidal rule was used to integrate theROC values. This method of calculating theROC is similar to the method described bySwets [17].

The performance on the samples used totrain a classifier is typically an optimisticestimate of the classifier’s performance on anew set of data [18]. To estimate the true errorrate, we used cross-validation to partition thedata into two subsets that are used to trainand test the data. In the first classificationproblem, we used 50 randomly selected parti-tions for which 70% of the data were used totrain the classifiers. A different classifierwas trained and tested on each partition.The mean classification rate and integratedROC on the testing subsets were used toevaluate the expected performance of aclassifier.

The second classification problem has amuch smaller set of data, so we performed10-fold cross-validation. The data set wasrandomly partitioned into ten sets, each ofwhich contained one positive and negativeexample that were used for testing theclassifier. Again, classifiers were trained andtested on each partition. The mean classifica-tion rate was used to evaluate the expectedperformance of a classifier. Because of thesmall size of the data set, the integrated ROCwas calculated once using the values of thelogit model for every test sample.

3.3. Experiments

In both classification problems, we per-formed experiments that examine the classifi-cation rate using each tortuosity by itself. Wealso performed experiments using all of the


Table 2Results for the problem of classifying blood vessel segments and blood vessel networks. The mean of thecross-validation classification rate and integrated ROC

Measure NetworksSegments

Classification rate Integrated ROC Classification rate Integrated ROC

Auto ManualManual Auto Manual Auto Manual Auto

91.3 0.897 0.955 6579.5 90t1 0.61 0.9179.3t2 82.7 0.875 0.915 70 85 0.79 0.8582.9t3 89.5 0.927 0.960 90 90 0.86 0.86

88.5 0.905 0.951 8080.4 95t4 0.83 0.9987.7 0.906 0.941 85 90t5 0.8481.2 0.8389.1 0.921 0.956 8082.1 90t6 0.81 0.91

82.0t7 88.1 0.914 0.944 85 90 0.88 0.91

All 91.585.6 0.935 0.970 95 90 0.95 0.86

tortuosity measures together. We appliedTukey’s method for multiple comparisons[19] to see if there were significant differ-ences in the classification rate of theclassifiers in the two classification problems(pB0.05).

4. Results

Table 2 summarizes the experimental re-sults for the two classification problems. Theexperiment name indicates whether the datawere manually extracted or automaticallyextracted and indicates which tortuositymeasures were used in the experiment. Forthe segment classification problem, the man-ually extracted data set exhibited significantdifferences between the following pairs oftortuosity measures: (t1, t3), (t1, t6), (t1, t7),(t2, t3), (t2, t6) and (t2, t7). All of the inter-actions in the automatically extracted dataset were significantly different, except thefollowing: (t3, t6), (t4, t5), (t4, t6), (t4, t7) and(t5, t7). For the network classification prob-lem, no significant differences were found

for either the manually or automatically ex-tracted data sets.

For most of the classifiers, the relativeintegrated ROC values mirrored the relativeclassification rates. An interesting exceptionis the integrated ROC for t1. For the seg-ment classification problem, the classifica-tion rate of the classifier using t1 to classifyautomatically extracted blood vessels wassignificantly higher than the rest of theclassifiers. However, the integrated ROC fort1 was lower than the integrated ROC for t3

and t6. This indicates that the classificationthreshold used in these experiments favorst1, but the t3 and t6 measures are robust fora wider range of classification thresholds.

In preliminary experiments, we examinedtortuosity measures that multiplied thelength-normalized measures by the averagevessel diameter. However, these measuresperformed slightly worse than their counter-parts. We thought these would improve theperformance of our tortuosity measuressince this increases the measure’s scale rela-tion without changing its tortuosity proper-ties.


5. Discussion

Our experimental results demonstrate thatthe proposed tortuosity measures can beused to effectively classify the tortuosity ofblood vessel segments and blood vessel net-works. In particular, these results show thatthe tortuosity measures can be used withautomatically extracted blood vessel seg-ments, which is important for the develop-ment of retinal image analysis tools.

5.1. Blood 6essel segment tortuosity

The better classification performance onthe automatically extracted data set can beattributed to the short, simple blood vesselsegments in this data set. While an entirevessel segment may be labeled tortuous,subsegments of the vessel may not be par-ticularly tortuous. These non-tortuous sub-segments will influence the overall tortuosityof the vessel segment, making it appear lesstortuous and thereby more difficult to clas-sify. The automatically extracted blood ves-sels are usually short enough to be entirelytortuous or non-tortuous.

This also explains why the t1 tortuositymeasure performed much better on the au-tomatically extracted data set. This measuredoes not accurately discriminate betweenlong segments because it does not analyzethe tortuosity along the segment. Since thetortuosity is more constant along shortersegments, this global measure performs well.

We can use the statistical analysis of theclassification rates to analyze the relativeimportance of the abstract tortuosity mea-sures. The only clear pattern for both ofthe data sets is that the t2 measure hadconsistently poor performance. In fact, t2

was significantly worse than t3 for bothdata sets. The only property not shared byt2 and t3 is response to scale, which sug-

gests that the response to scale is an impor-tant property. The fact that the differencebetween t2 and t3 was greater on the manu-ally extracted data set supports this conclu-sion.

5.2. Blood 6essel network tortuosity

The results for the vessel network classifi-cation problem follow the same pattern asthe results for the blood vessel classificationproblem. The classification performance wasbetter on the automatically extracted dataset and the t1 tortuosity measure performedmuch better on the automatically extracteddata set.

It is surprising that the performance alsoincreased for t4 and t5. These measureshave the compositionality property, so theblood vessel networks should have roughlythe same tortuosity measure for both datasets. An analysis of the data set indicatedthat the performance difference was due toa difference in the extraction process onone of the images. This particular imagewas tortuous, but only slightly so. The au-tomatically extracted segments includedsome exudate edges that were misclassifiedas blood vessel edges during extraction.These edges had a high curvature andthereby made the image appear more tortu-ous. The manually extracted segments didnot include these segments and thereby re-duced its tortuosity measure to a pointwhere the image appeared non-tortuous.

One other image was responsible formost of the remaining error for both datasets. This image is clearly tortuous, but ithas very small blood vessels. The segmenta-tion algorithm tends to smooth out verysmall blood vessels, so the tortuosity mea-sures for this image were low enough tomake it appear non-tortuous.


5.3. Blood 6essel extraction

The experimental results indicate that theutility of the tortuosity measures can be af-fected by the manner in which the bloodvessel segments are extracted. We have ob-served three different ways in which the ex-traction process influenced the performancein the classification tasks. While these influ-ences are specific to our data, they highlightfactors that will influence the performance oftortuosity measures on any data set.

First, the segmentation method can affectthe tortuosity of the extracted vessels. Oursegmentation algorithm breaks up vessels atpoints of very high curvature and tends tosmooth out very thin blood vessels. Theseissues are less problematic when calculatingthe tortuosity of blood vessel networks. Thebroken blood vessels can be reconnected andfactors that cause tortuosity will usually af-fect both large and small blood vessels.

The length of the extracted segments canalso affect the utility of the tortuosity mea-sure for classification. The shorter segmentsin the automatically extracted data sets is theprinciple reason for the better classificationresults for these data sets.

Finally, extracting blood vessel segmentsautomatically can effect the tortuosity mea-sures of blood vessel networks since they caninclude misclassified segments. Non-bloodvessel segments are often more tortuous thanblood vessel segments, so the tortuosity of ablood vessel network can be biased whenmisclassified segments are included. However,misclassified segments occur almost only inimages that contain a large number of abnor-mal objects such as lesions. This problem didnot seriously impact our results because theblood vessel classifier used to automaticallyextract the blood vessel segments is quiteaccurate. It has a 89.6% classification rate fornon-blood vessels on a data set containingimages with lesions [13].

6. Conclusions

Our experimental results demonstrate thatthe proposed tortuosity measures can be usedto classify the tortuosity of blood vessel seg-ments and blood vessel networks. We believethat automated tortuosity measures like thesewill play an important part of retinal imageanalysis tools. Automated measures are im-portant because they provide consistent mea-surements that are directly comparable andcan be used to detect changes in vasculartortuosity accurately.

Both classification problems used a coarse,two-class scale to evaluate the tortuositymeasures. A coarse scale was chosen for theexperiments to minimize the possible error ofthe hand labeling. One advantage of the pro-posed tortuosity measures is their ability tomake tortuosity measurements in a continu-ous fashion. However, additional experimentsare needed to evaluate whether the proposedtortuosity measures correctly predict tortuos-ity on a finer scale.

We have motivated our choice of tortuos-ity measures with abstract properties thatformalize ophthalmological intuitions con-cerning tortuosity measures. Abstract proper-ties provide a formal basis for comparingtortuosity measures, which enables us to ana-lyze the prospective performance of new tor-tuosity measures. Our experimental resultssuggest that response to scaling is an impor-tant property for tortuosity measures, whilethe compositionality properties did not exhib-ited discernible influence on the classificationperformance. The importance of the responseto scaling is particularly interesting becausethe results indicate that inverse scaling ispreferred, which is contrary to our ophthal-mological intuitions.

While our classification results do notstrongly support the use of one measure overthe others, the t3 measure appears the best


choice for a tortuosity measure. Both the t3

and t4 measures were closest to the ophthal-mologist’s notion of tortuosity, though the t3

measure performed slightly better on the clas-sification tasks. Classification with all of thetortuosity measures was better than t3, butnot in a statistically significant manner.

Because tortuosity measures are used toinfer information about disease severity orchange of disease over time, the reproducibil-ity of tortuosity measures is an importantproperty. We have not examined this issue inthis study, but the analysis in Smedby et al.[5] suggests that our measures will be veryreproducible. Smedby et al. [5] observe thatthe total curvature (t2) and distance factor(t1) measures are very reproducible across aseries of images of the same blood vessel. Allof the tortuosity measures that we have ex-amined are very similarly (if not identical) tot1 and t2.

Acknowledgements

This work was supported by NIH GrantNo. RO1LM05759. This work was per-formed in part at Sandia National Laborato-ries. Sandia is a multiprogram laboratoryoperated by Sandia corporation, a LockheedMartin Company, for the United States De-partment of Energy under Contract DE-AC04-94AL85000.

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Measurement and classiﬁcation of retinal vascular tortuositysumbaug/RetinalProjectPapers/Measurement … · Keywords:Tortuosity; Retina; Blood vessel; Automated measurement 1. Introduction