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Ray tracing model for the estimation of power spectral properties in laser Doppler velocimetry of retinal vessels and its potential application to retinal vessel oximetry

Benno L. Petrig and Lysianne Follonier

Open Access Open Access

Abstract

A new model based on ray tracing was developed to estimate power spectral properties in laser Doppler velocimetry of retinal vessels and to predict the effects of laser beam size and eccentricity as well as absorption of laser light by oxygenated and reduced hemoglobin. We describe the model and show that it correctly converges to the traditional rectangular shape of the Doppler shift power spectrum, given the same assumptions, and that reduced beam size and eccentric alignment cause marked alterations in this shape. The changes in the detected total power of the Doppler-shifted light due to light scattering and absorption by blood can also be quantified with this model and may be used to determine the oxygen saturation in retinal arteries and veins. The potential of this approach is that it uses direct measurements of Doppler signals originating from moving red blood cells. This may open new avenues for retinal vessel oximetry.

©2005 Optical Society of America

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Figures (5)
Fig. 1.
Fig. 1. (a) Cartesian coordinate system for model calculations with origin at the retina. (b) Vessel axis is on x-axis with blood velocity vector pointing in the direction of the positive x-axis. The y-axis is tangential to the retina, z-axis goes through center of eye. Laser beam axis is in the direction of the negative z-axis intersecting the y-axis at eccentricity y 0 from the vessel axis. Detector axis lies in a plane parallel to the xz-plane at y 0 (dashed rhombus) and makes an angle α 1 with the laser axis.

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Fig. 2.
Fig. 2. Model calculations of Doppler signal power spectral shapes in a round vessel (radius R = 50μm). Laser illumination (670 nm) is either uniform (traditional LDV, dashed line) or has a Gaussian shape with beam radii w = 5, 2, 1, 0.5, 0.2 and 0.1R. Laser beam and vessel axes are in the same plane, perpendicular to each other. Scattered beam angle: α 1 = 10°.

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Fig. 3.
Fig. 3. Power spectral shapes in a round vessel (R = 50μm) for a truncated Gaussian beam (radius w = 0.2R) as a function of eccentricity y 0 between laser beam and vessel axes. Scattered beam angle: α 1 = 10°.

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Fig. 4.
Fig. 4. Power spectral shapes in a round vessel (R = 50μm) for a truncated Gaussian beam (radius w = 0.2R) as a function of wavelength (532, 569, 670, 810 nm), with or without taking into account wavelength-dependent absorption of light by oxy- or deoxyhemoglobin. Laser beam and vessel axes intersect at the origin. Scattered beam angle: α 1 = 10°.

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Fig. 5.
Fig. 5. Specific absorption coefficient spectrum of oxy- and deoxyhemoglobin. The four wavelengths calculated in our model are shown as vertical dash-dotted lines, two of them (569 and 810 nm) are isobestic points. Data are taken from van Assendelft [13].

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Equations (15)

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v ( r ) = ( 1 − r 2 R 2 ) V max ,
I 1 ( ρ ) = I 0 exp ( − 2 ρ 2 w 2 ) ,
f ( r ) = ( 1 2 π ) [ v ( r ) · ( k s − k i ) ] ,
f ( r ) = v ( r ) sin ( α 1 ) n λ ,
I sc ( r ) = γ ∫ x min x max ∫ y min ( x ) y max ( x ) I 1 x y l arc y r d y d x ,
I sc ( r ) = γ ∫ x min x max ∫ y min ( x ) y max ( x ) I 1 x y exp [ − μ a ( λ ) d ( y ) ] l arc y r y d x ,
d ( y ) = { 2 [ ( R 2 − y 2 ) 1 2 − ( r 2 − y 2 ) 1 2 ] z ≥ 0 , 2 [ ( R 2 − y 2 ) 1 2 + ( r 2 − y 2 ) 1 2 ] z < 0 .
I sc ( f ) = λ R 2 2 n sin ( α 1 ) V max I sc ( r ) r .
DSPS ( f ) = β ( λ ) 2 SS lo I lo I sc ( f ) ,
P Hb O 2 . sh . 532 P HbO 2 . sh . 810 = k 0 ( ≈ 0.125 ) ,
P Hb . sh . 532 P Hb . sh . 810 = k 1 ( ≈ 0.25 ) .
P sh . 532 = α P Hb O 2 . sh . 532 + ( 1 − α ) P Hb . sh . 532 .
P Hb O 2 . sh . 810 = P Hb . sh . 810 = P sh . 810
SO 2 =α* 100 % = k 1 P sh . 810 − P sh . 532 ( k 1 − k 0 ) P sh . 810 * 100 % ,
SO 2 =α* 100 % = k 1 P sh . 810 − k 2 P sh . 532 ( k 1 − k 0 ) P sh . 810 * 100 % ,
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