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Measurement and Classification of Retinal Vascular Tortuosity

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Measurement and Classification of Retinal Vascular Tortuosity International Journal of Medical Informatics 53 (1999) 239–252 Measurement and classification of retinal vascular tortuosity William E. Hart a,*, Michael Goldbaum b, Brad Coˆte´ c, Paul Kube d, Mark R. Nelson e a Department of Applied and Numerical Mathematics, Sandia National Laboratories, Alburquerque, NM 87185, USA b Department of Ophthalmology, Uni6ersity of California, San Diego, CA, USA c The Registry, Inc., San Diego, CA, USA d 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 diagnostic tools. We describe a suite of automated tortuosity measures for blood vessel segments extracted from RGB retinal images. The tortuosity measures were evaluated in two classification tasks: (1) classifying the tortuosity of blood vessel segments and (2) classifying the tortuosity of blood vessel networks. These tortuosity measures were able to achieve a classification rate of 91% for the first problem and 95% on the second problem, which confirms that they capture much of the ophthalmologists’ notion of tortuosity. Finally, we discuss how the accuracy of these measures can be influence by the method used to extract the blood vessel segments. © 1999 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Tortuosity; Retina; Blood vessel; Automated measurement 1. Introduction because of longitudinal stretching. The tortu- osity may be focal, occurring only in a small Normal retinal blood vessels are straight or region of retinal blood vessels, or it may gently curved. In some diseases, the blood involve the entire retinal vascular tree. Fig. 1 vessels become tortuous, i.e. they become di- shows images with tortuous and non-tortu- lated and take on a serpentine path. The ous blood vessels. dilation is caused by radial stretching of the Many disease classes produce tortuosity, blood vessel and the serpentine path occurs including high blood flow, angiogenesis and blood vessel congestion. High blood flow * Corresponding author. E-mail: [email protected]. may occur locally in arteriovenous anastamo- 1386-5056/99/$ - see front matter © 1999 Elsevier Science Ireland Ltd. All rights reserved. PII: S1386-5056(98)00163-4 240 W.E. Hart et al. / International Journal of Medical Informatics 53 (1999) 239–252 Fig. 1. Images with (a) tortuous and (b) non-tortuous blood vessel segments. sis or systematically in high cardiac output between observers and with the same ob- such as accompanies anemia. In response to server at different times. Furthermore, mea- ischemia or inflammation, new blood vessels surements on such a gross scale make grow, often in a destructive manner. The changes in vascular tortuosity difficult to angiogenic process commonly causes dilation discern. and tortuosity of retinal blood vessels prior A quantitative tortuosity measurement was to the onset of the neovascularization. first described by Lotmar, Freiburghaus and Venous congestion can result from the occlu- Bracher [1] and slightly extended by Bracher sion of the central retinal vein or a branch [2]. This measurement involves the manual retinal vein. The resulting elevated intravas- selection of points on fundus photographs cular pressure results in tortuosity and dila- that subdivide the tortuous vessel into a se- tion in the blocked vein. ries of single arcs. Fig. 2 illustrates the mea- Information about disease severity or change of disease with time may be inferred by measuring the tortuosity of the blood vessel network. Consequently, there is a benefit in measuring tortuosity in a consis- tent, repeatable fashion. Vascular tortuosity measurements are commonly performed by human observers using a qualitative scale (e.g. mild, moderate, severe and extreme). Fig. 2. Illustration of the subdivision used to estimate Variability in grading can result because the blood vessel tortuosity with the relative length varia- boundaries between the grades may differ tion. W.E. Hart et al. / International Journal of Medical Informatics 53 (1999) 239–252 241 surements they use to calculate the vessel’s of tortuosity. To assess the relative utility of tortuosity. Their method decomposes the ves- these measures, they were used to classify sel into a series of circular arcs, for which the blood vessel segments and blood vessel net- chord lengths li and arrow heights hi are works. The segments used in these classifica- measured. The tortuosity is measured as the tion experiments were extracted manually relati6e length 6ariation and automatically and we discuss how the tortuosity measures can be influenced by L−l 8 h 2 : % i , properties of the method used to extract the l 3 i li vessel segments. The classification rate was as where L is the length of the blood vessel and high as 91.5% for blood vessel segments and l is the chord length. The approximation is 95% for blood vessel networks. While no derived using a sinusoidal model of a blood single tortuosity measure was clearly supe- vessel. This method was used by Kylstra et rior, the experiments recommend a total al. [3] to measure the effects of transmural squared curvature measure. pressure on the tortuosity of latex tubing. Given the increased availability of digitized fundus photographs, automated tortuosity 2. Tortuosity measures measurements are now feasible. Kaupp et al. [4] have reported unpublished results of an 2.1. Abstract properties automated tortuosity measurement that uses a Fourier analysis of the perpendicular along We begin by discussing several properties the blood vessel. Smedby et al. [5] describe of tortuosity measures that are motivated by five tortuosity measures used to measure tor- the ophthalmologist’s notion of tortuosity. A tuosity in femoral arteries. Included are sev- tortuosity measure that has many of the eral measures of the integral curvature along properties described below should provide the blood vessel, the number of inflection good predictive performance in our classifica- points of the vessel and the fraction of the tion tasks, thereby matching an ophthalmolo- vessel that has high curvature. Their experi- gist’s measure of tortuosity. We will discuss ments examine properties of these measures the following properties of tortuosity mea- like reproducibility and scalability. Ca- sures: invariance to translation and rotation, powski, Kylstra and Freedmen [6] describe a response to scaling and compositionality of measure of blood vessel tortuosity based on vessels and vessel networks. These properties spatial frequencies. Zhou et al. [7] have also are defined for parametrized differentiable described a method for distinguishing tortu- curves C=(x(t), y(t)), with t in an interval ous and nontortuous blood vessels in an- [t0, t1]. We model retinal blood vessels with giograms. In a preliminary abstract, we have such curves. A tortuosity measure t takes a proposed a tortuosity measure based on the curve C as its argument and returns a real integral curvature along a blood vessel [8]. number. In this paper we describe tortuosity mea- sures that are used to measure the tortuosity 2.1.1. In6ariance to translation and rotation of retinal blood vessels as well as the retinal Suppose a vessel with tortuosity t were blood vessel network. We motivate our tortu- moved on the retina without changing its osity measures by analyzing abstract proper- shape or size. What should be the tortuosity ties of measures based on medical intuitions of the resulting vessel? 242 W.E. Hart et al. / International Journal of Medical Informatics 53 (1999) 239–252 While ophthalmologists’ judgments of ves- appears that if a vessel segment with a given sel tortuosity do seem to incorporate the tortuosity were extended with a segment of relative curvature of other vessels on the fun- the same tortuosity, the resulting vessel dus, we believe their judgment is largely inde- would have the same tortuosity as either of pendent of the location or orientation of the its segments. That is, a vessel doesn’t increase vessels. Our measures assume that vessel tor- (or decrease) its tortuosity by having the tuosity is not dependent on the location of same serpentine pattern extended at greater orientation of the vessel. length. We formalize these intuitions as follows. 2.1.2. Response to scaling We write C=C1 C2 and say curve C has Suppose a vessel with tortuosity t was constituent segments C1 and C2. Whenever enlarged uniformly by some scale factor g\ C=(x(t), y(t)) with t [t0, t1], C1 =(x(t), 1. What should be the tortuosity of the scaled y(t)) with t [t0, t2] and C2 =(x(t), y(t)) with vessel? t [t2, t1], for some t2 [t0, t1]. Now suppose t 5t Ophthalmologists do not seem to have un- that (C1) (C2). We say that a tortuosity equivocal intuitions about this question, measure has the property of compositionality though they believe that a vessel’s tortuosity when, for any such C1, C2, C3, should be roughly invariant to scaling. How- t(C )5t(C)5t(C ) (1) ever, it is clear that if scale does affect the 1 2 tortuosity then it does so in a multiplicative with the equality holding if and only if t t manner. That is, if curve C=(x(t), y(t)) has (C1)= (C2). tortuosity t(C), curve C%=(gx(t), gy(t)) has The property of compositionality provides tortuosity b(g)·t(C) for some function b. constraints on how to compute the tortuosity If tortuosity is invariant to scaling, b(g) 1; of a vessel, given the tortuosities of its con- and if tortuosity is inversely related to scale, stituent segments. This computation will be b(g)B1 when g\1. We examine tortuosity necessary in an automated system that ex- measures that exhibit various tortuosity scale tracts vessel segments and not whole vessels factors b(g) and investigate their effect on from fundus images.
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