A novel method for the automatic evaluation of retinal vessel tortuosity

1
A novel method for the automatic evaluation of
retinal vessel tortuosity
Enrico Grisan, Marco Foracchia and Alfredo
Ruggeri
Abstract
Tortuosity is among the first alterations in
retinal vessel network to appear in many
retinopathies. Automatic evaluation of retinal
vessel tortuosity is thus a valuable tool for
early detection of vascular suffering. Quite a
few techniques for its measurement and
classification have been proposed, but they do
not always match the clinical concept of
tortuosity. This justifies the need for a new
definition, able to express in mathematical terms
the tortuosity as perceived by ophthalmologists.
We propose here a new algorithm for the
evaluation of tortuosity in vessels extracted
from digital fundus images. It is based on the
partitioning of each vessel in segments of
constant-sign curvature and on the combination
between the number of such segments and their
curvature values. This algorithm has been
compared with the other tortuosity measures on a
set of 20 vessels from 10 different images. These
vessels had been preliminarily ordered by an
expert ophthalmologist in order of increasing
perceived tortuosity. The proposed algorithm
proved to be the best one with regards to
arterial tortuosity and among the best for vein
tortuosity evaluation.
Composition
Introduction
Proposed Tortuosity Measure
Adjacent continuous curves s1 and s2, the
combination of the two is In 1, an
intuitive principle was proposed for the
tortuosity of the composition

• Integrating information
• Number of times a vessel changes the curvature
  sign (twist)
• Amplitude of each twist
• 1 Being Cs(l) the curvature of segment s,
  described by the curvilinear coordinate l
  belonging to the domain D, each vessel s is
  partitioned in a set of n subsegments si,
  i1,..., n such that
• This represents an hysteretic thresholding of the
  curvature (Fig. 3)
• Each subsegment has a quasi-constant curvature
  sign (the quasi given by the hysteresis)

• Many diseases have the retina as target organ,
  and some are only recognizable by changes in the
  vascular network of the retina.
• One of the first changes is the increase in
  vessel tortuosity
• Tortuosity measure has to match the clinical
  perception of ophthalmologists
• Understand the factors that influence the
  classification of a vascular structure as
  tortuous or non-tortuous.
• In retinal images, straight vessels but also long
  vessels presenting a smooth semi-circular shape
  are considered as non-tortuous.
• Previously proposed methods (see 1 for a
  review) failed in differentiating the tortuosity
  of structures that visually appeared to be very
  different in tortuosity, as it will be shown.

• Fig. 2 top panel, shows that this statement can
  not be accepted in conjunction with the principle
  of invariance with respect to rotation and scale
  
• connecting three non-tortuous curves yields an
  undoubtedly tortuous curve.
• New composition property, such that a vessel s,
  combination of various segments si, will not have
  tortuosity measure less than any of its composing
  parts

Methods
Available Data
Figure 3 Vessel subsegment evaluation, based on
the hyteretic thresholding of the curvature
Figure 2 Tortuosity measures counterexamples (see
text)
A set of 20 vessels from 10 different retinal
images was used. Images were acquired with a
fundus camera with a 50o field of view and then
digitized vessel segments were automatically
extracted by a previously developed tracking
algorithm 5, and sorted by increasing
tortuosity by an expert ophthalmologist (arteries
and veins separately ) A vessel is a continous
curve in a two dimensional space, and it can be
described by a sampled version of it.
Modulation

• For two vessels having twists (changes in
  curvature sign) with the same amplitude (maximum
  distance of the curve from the underlying chord),
  the difference in tortuosity varies with the
  number of twists frequency modulation.
• For two vessels with the same number of twists
  (with the same frequency), the difference in
  tortuosity depends on the difference in amplitude
  of the twists amplitude modulation

Results and discussion
Available Tortuosity Measures

• The proposed tortuosity measure
• Is invariant to rotation and translation
• Composition is fulfilled through the summation
• Amplitude modulation is accounted for with Ri
• Frequency modulation is accounted for through
  vessel subdivision (implicitly), and through the
  multiplication by n-1 (explicitly)
• Division by L makes the tortuosity measures
  indipendent from scaling.
• The proposed measures achieved a correlation of
  0.870 for the artey vessel set (best among all
  the measures proposed in 1,2,4)
• The proposed measures achieved a correlation of
  0.870 for the vein vessel set (third best among
  all the measures proposed in 1,2,4)

• Sampling of a vessel may lead to a very sparse
  vessel description
• Poor information on vessel direction and its
  derivatives
• Noisy sample direction information
• Cubic smoothing spline fit
• describes the vessel between sampling points,
  where no data are present.
• give a C1 (at least) representation of the
  vessel (physiologically sound description)

Arc Length over length ratio

• Ratio between its length and the length of the
  underlying chord
• The greater the value of the ratio, the more
  distant the vessel is from a straight line, i.e.,
  tortuous 123
• Being the surface of the retina close to a
  semi-sphere, the non tortuous paradigm should be
  the circle arc
• Fig. 2 (top and middle panel) shows that two
  vessels with the same arc/chord ratio may have
  very different perceived tortuosity

Measure involving curvature

• Various positive functions of the curvature 1
  curvature should be a measure of the variability
  of vessel direction.
• The curve in Fig. 2 middle panel integral of the
  absolute value of curvature is p
• The curve in Fig. 2 bottom panel integral of the
  absolute value of curvature is p/2
• The bottom panel curve has greater perceived
  tortuosity
• Composition of straight lines and arcs
  dramatically changes tortuosity perception
• Changes in curvature sign are not taken into
  account
• Integral value depends upon the integration domain

Acknowledgements
This work was partially supported by a research
grant form Nidek Technologies, Italy. The authors
would like to thank Prof. S. Piermarocchi and
colleagues at the Department of Ophthalmology,
University of Padova, for providing images and
clinical advice. M. Foracchia is now with M2
Scientific Computing, Italy.
Figure 1 Vessel centerline and borders sample
(left panel), and vessel samples interpolation
through cubic spline (right panel)
Tortuosity Properties
Affine Transformation
Bibliography

• Geographical position in the retina does not
  affect tortuosity perception translation
  invariant
• Vessel orientation does not affect tortuosity
  perception rotation invariant
• Scale does not seem to affect tortuosity
  perception, but this is controversial. The
  evaluation of a single vessel tortuosity might be
  considered invariant to scaling

1 W. Hart et al., Int. J. Med. Inf., 53,
239-252, 1999 2 C. Heneghan et al., Med. Imag.
An., 6, 407-429, 2002 3 M. E. Martinez-Perez et
al., IEEE Trans. Biom. Eng., 20, 1193-1200,
2002 4 K. V. Chandrinos et al., Proc.
MEDICON98, 1998 5 M. Foracchia et al., CAFIA
2001, 15, 2001
Mean direction angle change

• Average of difference in vessel directions among
  samples within a distance window 4
• Sensitivity to noise
• Difference in direction fails to account for
  changes in convexity