Albrecht v. Graefes Arch. klin. exp. Oplithal. 211, 49-57 (1979)

Graefes Archiv flJr klinische und experimentelle

Ophthalmologie 9 by Springer-Verlag 1979

Measurement of Vessel Tortuosity on Fundus Photographs* W. Lotmar 1 **, A. Freiburghaus 2, and D. Bracher 2 1 Eye Clinic (Director: Prof. P. Niesel), 2 Pediatric Clinic (Director: Prof. E. Rossi) of the University of Berne, Switzerland

Summary. Quantitative measurement of vessel tortuosity and its variation on fundus photographs is a sensitive means of obtaining information about the course of an asphyctic event in newborns, virtually independent of bias produced by the photographic process. We subdivide a tortuous vessel into single arcs and measure the chord length and the arrow height (Pfeilh6he) of every arc on a projected image of the film. From these figures, a fairly accurate value for the relative length increase of the arc, as compared with the chord, can be derived by a simple approximation formula [Eq. (5)]. It is shown that neglect of the third dimension, not visible on an ordinary photograph, entails only a small error. Trained observers achieve results reproducible to about 1% in relative length variation. Zusammenfassung. Die quantitative Messung abnormaler GefN3windungen und ihrer Ver/inderungen auf Fundusbildern ist eine empfindliche Methode zur Erlangung yon Information fiber den Verlauf eines asphyktischen Ereignisses bei Neugeborenen. Sie ist praktisch unabh/ingig yon m6glichen Verffilschungen durch den photographischen Prozel3. Wir teilen ein gewundenes Geffil3 in Einzelb6gen auf und messen an jedem derselben die Sehnenlfinge und Pfeilh6he im projizierten Bild. Aus diesen Werten kann mit Hilfe einer einfachen Nfiherungsformel [G1. (5)] ein ziemlich genauer Wert der prozentischen Lfingenzunahme des Bogens gegenfiber der Sehne gewonnen werden. Es wird gezeigt, dab die Vernachlfissigung der dritten im Bild nicht sichtbaren Raumdimension dabei nur einen geringen Fehler verursacht. Im Messen gefibte Personen erreichen Ergebnisse, die auf etwa 1% in relativer L~ingenfinderung reproduzierbar sin&

Introduction

Abnormal tortuosity of retinal vessels has been observed, for example, in connection with hypertension (Kagan et al., 1967), coarctation (Eisalo et al., 1970) * Supported by the Swiss National ScienceFoundation ** Address for offprint requests: Dr. W. Lotmar, Augenklinik des Inselspitals, Freiburgstrasse 8, CH-3010 Bern, Switzerland

0065-6100/79/0211/0049/$01.80

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W, L o t m a r et al.

and other cardiac malformations (Crowe et al., 1969), and retrolental fibroplasia (Baum, 1971). We have shown that in the newborn marked changes in tortuosity occurring within a few hours or days after an asphyctic event can be photographically documented (Bracher et al., 1975). The study of tortuosity is of particular interest for the following reasons: Lemmingson (1972) has shown that the width of retinal vessels appearing in conventional fundus pictures effectively corresponds to the diameter of the erythrocyte column. Hence, changes in vessel width may be due to plasma skimming, to swelling of the endothelial cells, or to changes in muscular tone, whereas changes in tortuosity are very likely caused by variation of muscular tone alone and will therefore provide a clearer insigth into the mechanisms of regulation and the pathophysiology involved. Moreover, observable changes in tortuosity will hardly be affected by any deficiencies in the photographic process, in contrast to the situation when one attempts to measure vessel width (Bracher et al., 1979). The present paper is concerned with the principles of a new method for measurement and evaluation of tortuosity. Application to newborns under physiological and pathological conditions will be reported elsewhere (Thalmann et al., in preparation).

Theoretical Basis

To begin with, it should be noted that pathological vessel tortuosity as observed in fundus photographs comprises only two dimensions of the phenomenon, whereas in reality it is likely to be a three-dimensional phenomenon. We shall show, however, that information obtained from two-dimensional pictures covers the greater part of the three-dimensional case. We shall therefore begin with the former. Kagan et al. (1967) used a curvometer to measure vessel length on fundus photographs. They projected the pictures at about fiftyfold magnification and measured vessel length between two concentric circles differing in radius by 100 mm. According to our experience, accuracy of measurement with a confidence range of 95% is of the order of 2% under these conditions.

Fig. 1. Schematic subdivision of a tortuous vessel into circular arcs

A

B

Fig. 2, Example of a curved vessel 1% longer than its chord between points A and B

Vessel Tortuosity

51

Fig. 3. Relation between arc and chord

A m~re sensitive method for determining variations in vessel length on fundus photographs is to subdivide a tortuous vessel into a series of single arcs and to measure the chords and arrow heights (Pfeilh6hen) of these arcs, as shown in Fig. 1. F r o m these data, absolute or relative length variations can be obtained by using a simple approximation formula. To illustrate the sensitivity of this method, we refer to Fig. 2, in which a curved vessel that is 1% longer than its chord between points A and B is schematically shown. We have found that in this case the arrow height can still be measured on a real photograph by a trained observer to an uncertainty of typically _+10%, leading to an uncertainty in length variation of only 2%. In the retina of a newborn, this corresponds to about 2.5 p~m and is better by an order of magnitude than that which can be achieved with a curvometer. Let us now consider the derivation of the approximation formula mentioned above. We assume that the tortuous vessel consists of a series of circular arcs of individual curvature, as shown in Fig. 1, each characterized by its chord li and arrow height h~. The sequence of chords is thought to represent the vessel in its ' n o n d i s t o r t e d ' form. The length of a single arc as compared with its chord is obtained as follows: F r o m Fig. 3 we have

(~)2= r2-(r-h)2=h(2r

-

h). (1)

Neglecting h against 2r we obtain

12~-8hr.

(2)

On the other hand, the arc length L

l = r. arc sin ~rr"

(3)

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W. L o t m a r et al.

a

Fig. 4. Equivalence of two modes of measurement

We develop arc sin into a series of which we retain only the first two members: X3

arc sin x = x + ~ - + .... If we apply this to (3) and replace r by L 2-2

I2/8h from

(2) we get

1 8h 2 ~ 61

i.e., the length difference 8 h2

L-I=Al~-3 1"

(4)

The relative length variation then

I-3

(5)

In spite of the neglects we have made, approximation by this formula is quite good: even for an arc length one-third of a full circle, where the arc is 21% longer than the chord and h/l=0.29, the error in Al/l from (5) amounts only to + 1.5%. In practice, measurement of h and l on a vessel as assumed in Fig. 4b would not be very convenient. It is preferable to measure just one value of h and l as shown in Fig. 4a. This is quite legitimate, however, since it can easily be shown that the result is indeed the same for hi=h3, and that it will differ only slightly for hi +h3. We tested Eq. (5) on a two-dimensional model consisting of a thin plastic tube filled with Indian ink, whose length between two reference points could be varied by known amounts. The tube lay on a horizontal board and was photographed from above on 35-mm film. These pictures were evaluated on a distortion-free tenfold desk projector (Fig. 10) 1. The image was projected 1

Instrument from Projectina A G , 9435 Heerbrugg, Switzerland

Vessel Tortuosity

% ~25.

53

: douse s,~arcs arcs

/

t ~

15" ~10-

ib

~s actual

2b

g'/.

length increase

Fig. 6. Relation of actual and measured length increase for a planar plastic tube model

Fig. 7. Sine curve and corresponding circular arc. Length difference is 2.9%

onto a drawing paper screen containing a pencil line which was brought to a position tangent to the tube image as in Fig. 4a; h and l were measured with a transparent scale. Seven of the tube configurations were single and five were double arcs, as exemplified in Fig. 5. The results are shown in Fig. 6. The values found by application of Eq. (5) do not systematically deviate from true values up to 25% length increase. In view of the many assumptions and approximations made, however, it is no surprise that some spread is present. It is our experience that the form of tortuous vessels tends to be sinusoidal rather than a sequence of circular arcs. This can also be seen on the plastic tube in Fig. 5. We may therefore ask whether it would not be a still better approximation to start with a model of sine form. Apart from the fact that mathematical treatment is then more complicated, such a refinement appears to be hardly worthwhile since the lenghts of the sine line and the corresponding circle arc (see Fig. 7) differ only by 2.9%. When the arc is flatter, the difference is even less. Moreover, the results obtained with the plastic tube (Fig. 6) show that Eq. (5) obviously works quite well, irrespective of the exact curve form.

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W. Lotmar et al.

Fig. 5. A planar plastic tube model in two different tortuous modes Fig. 8. An elongated helical steel spring and its orthogonal sinusoidai shadow Fig. 9. Samples of tortuous vessels from fundus photographs of newborns as used for evaluation Fig. 10. Profile projector with fine-grained small wax layer screen at center of screen

Vessel Tortuosity

55

Three-Dimensional Case Experience with adults indicates that when abnormal tortuosity of vessels occurs, the displacements tend to lie within the plane of the retina. In this case, fundus pictures show the true length of the vessel. Let us suppose, however, in order to arrive at an upper limit of the possible error, that a vessel has assumed the shape of a regular elongated spiral spring (Fig. 8). What is the ratio between its real length and its projection as seen on the photograph? The projection is a sinusoidal line whose length can be calculated. The ratio depends on the pitch-to-diameter ratio of the spiral. For infinitely small pitch, its value approches 1.57 and decreases with relative elongation. When pitch is threefold the diameter, the ratio is 1.16, that is, the vessel length increase as calculated from Eq. (5) on the basis of a two-dimensional picture would be in error by not more than 16%. In most of the tortuous vessels of our material from newborns, the pitch-to-diameter ratio is rather more than 3 (in the four samples in Fig. 9, it varies between 4.5 and 7.3). Moreover, the picture show that, as a rule, the light reflexes on the arteries are visible all along their length with very little if any variation of intensity. This is a quite sensitive criterion for orientation of the vessels at practically right angles to the illuminating rays of the fundus camera, as can easily be verified by holding a steel wire some distance in front of the camera. In short, two-dimensional measurement of tortuosity will for the greater part result in length increase values close to the real ones.

Method of Measurement and Results

In order to evaluate intra- and interobserver reproducibility in the assessment of chord length and arrow height, we have used four fundus pictures of newborns, as shown in Fig. 9. On each picture, an arc of a tortuous artery as indicated was measured by six observers. We used a desk profile projector of tenfold magnification whose stage with the film could be displaced by micrometer screws (Fig. 10). Its screen, 18 cm in diameter, consisted of a glass .plate covered by translucent drawing paper with a 2.5x7.5 cm wax layer screen (Lotmar, 1954) at its center. Screens of this kind are especially fine-grained. It contained two fine marker lines at right angles, each parallel to one of the stage movements. The arterial arc was brought with its ' valleys' to a position tangent to the horizontal line (base line), as shown in Fig. 9, and the arrow height was measured by displacing the stage until the base line was tangent to the inner ' hill' border of the vessel. Chord length was measured by displacement between positions where first one and then the other tangent point of the valleys came to lie under the vertical marker line. Each observer had to assess chord length twice and arrow height four times, these six values constituting a 'single determination.' The film was then slightly displaced at random and a second determination made. For every determination, we took the mean of the chord lengths and used Eq. (5) to calculate the relative length increase for the four values of arrow height. Figure 11 shows the mean

W. Lotmar et al.

56

'l

b

c

d

e

I

5-

3 Number of arter

Fig. 11. Relative length increase measured twice by 6 observers (a) to 0O on 4 arteries as shown in Fig. 9 (numbers corresponding). Vertical lines are standard deviations

values and their standard deviations as determined by the six observers (two determinations each). Observers a and b were well trained, whereas observers c - f were new at this kind of task.

Discussion

The last statement is reflected in Fig. 11 insofar as agreement between the first two observers tends to be somewhat closer than with the other four. However, since this is only a moderate effect, it may be expected that relatively little training would suffice to improve the performance of untrained persons. When only the figures of the trained observers are considered, it appears that relative length variations of vessels within the range of 5-13% can be assessed to about 1% of uncertainty. Reproducibility achieved by one and the same observer may even be better. It should be remembered that the fundus pictures used for this study were of low contrast and partly of moderate definition, as shown in Fig. 9, since we were mainly interested in newborns. The reasons for our choosing low-contrast photographic material are given in Bracher etal. (1979).

Conclusions

The method for determining length variations of tortuous retinal vessels on fundus photographs by measuring the chord lengths and arrow heights (Pfeilh6hen) of single arcs is relatively easy and accurate. Use of a tenfold projector with a fine-grain screen is to be recommended. A simple formula for calculating

Vessel Tortuosity

57

a b s o l u t e o r relative l e n g t h c h a n g e f r o m these m e a s u r e m e n t s c a n b e d e r i v e d [Eqs. (4) a n d (5)]. A l t h o u g h the results, strictly s p e a k i n g , a p p l y to the t w o d i m e n s i o n a l p r o j e c t i o n o f t h e vessel o n t o the p i c t u r e p l a n e , n e g l e c t o f the t h i r d d i m e n s i o n e n t a i l s o n l y m i n o r errors.

References Baum, J.D. : Retinal artery tortuosity in ex-premature infants. Arch. Dis. Child. 46, 247-252 (1971) Bracher, D., Lotmar, W., Bossi, E., Vassella, F.: Funduso photography in neonates, an approach to obtain indirect information on cerebral blood flow. Helv. Paediatr. Acta 30, 473-486 (1975) Bracher, D., Dozzi, M., Lotmar, W. : Measurement of vessel width on fundus photographs. Albrecht v. Graefes Arch. klin. Ophthal. 211, 35-48 (1979) Crowe, R.J., Kohner, E.M., Owen, S.J., Robinson, D.M. : The retinal vessels in congenital cyanotic heart disease. Med. Biol. 19, 9549 (1969) Eisalo, A., Raitta, C., Kala, R., Halonen, P.J.: Fluorescence angiography of the fundus vessels in aortic coarctation. Br. Heart J. 32, 71-75 (1970) Kagan, A., Aurell, E., Tibblin, G. : Signs in the fundus oculi and arterial hypertension. Bull. WHO 36, 231-241 (1967) Lemmingson, W. : Vitalmikroskopische Untersuchnngen zur Morphologie und Pathogenese der experimentellen O2-Sch~idigung der Retina. Adv. Ophthalmol. 25, 240 322 (1972) Lotmar, W. : Translucent wax layer screens. Microtecnic 8, 75-79 (1954) Received December 18, 1978

Measurement of vessel tortuosity on fundus photographs.

Albrecht v. Graefes Arch. klin. exp. Oplithal. 211, 49-57 (1979) Graefes Archiv flJr klinische und experimentelle Ophthalmologie 9 by Springer-Verla...
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