Friday, December 7, 2018

Imaging and Velocimetry of the Human Retinal Circulation with Color Doppler Optical Coherence Tomography

Page 394 of Intermediate Physics for Medicine and Biology.
Page 394 of IPMB.
Section 14.7 of Intermediate Physics for Medicine and Biology discuses optical coherence tomography (OCT). Russ Hobbie and I write
Optical range measurements using the time delay of reflected or backscattered light from pulses of a few femtosecond (10-15 s) duration can be used to produce images similar to those of ultrasound…Since it is difficult to measure time intervals that short, most measurements are done using interference properties of the light. Optical coherence tomography is conceptually similar to range measurements but uses interference measurements…It is widely used in ophthalmology….
Intermediate Physics for Medicine and Biology Fig. 14.15.
Fig. 14.15 of IPMB.
The basic apparatus [for OCT] is shown in Fig. 14.15….The light pulse travels over an optical fiber to a 50/50 beam splitter. Part travels to the sample, where it is reflected back to the 50/50 coupler and then to the detector. The other half of the light goes to the reference mirror, where it is also reflected back to the detector. Changing the position of the reference mirror changes the depth of the image plane in the sample….

Intermediate Physics for Medicine and Biology Fig. 14.17.
Fig. 14.17 of IPMB.
It is possible to make many kinds of images. Fig. 14.17 shows the parabolic velocity profile of blood flowing in a retinal blood vessel 176 μm diameter. It was obtained by measuring the Doppler shift in light scattered from moving blood cells.
Yazdanfar et al. (2000).
Yazdanfar et al. (2000).
Figure 14.17 is from the paper “Imaging and Velocimetry of the Human Retinal Circulation with Color Doppler Optical Coherence Tomography,” by Siavash Yazdanfar, Andrew Rollins, and Joseph Izatt (Optical Letters, Volume 25, Pages 1448–1450, 2000). [For some reason Russ and I did not include the year in our citation—another item for the errata].
Abstract: Noninvasive monitoring of blood flow in retinal microcirculation may elucidate the progression and treatment of ocular disorders, including diabetic retinopathy, age-related macular degeneration, and glaucoma. Color Doppler optical coherence tomography (CDOCT) is a technique that allows simultaneous micrometer-scale resolution cross-sectional imaging of tissue microstructure and blood flow in living tissues. CDOCT is demonstrated for the first time in living human subjects for bidirectional blood-flow mapping of retinal vasculature.
I like Fig 14.17 because it combines ideas from Chapters 1, 11, 13, and 14 of IPMB. It also highlights the excellent spatial resolution you can obtain with OCT.

The illustration below shows the geometry associated with Fig. 14.17. The light is reflected by blood cells moving at speed v, causing a Doppler shift in its frequency. By adjusting the reference mirror, different depths are selected. The vessel makes an angle θ relative to the incident light. As the depth is scanned across the vessel, the Doppler shift determines the blood flow profile.
The geometry associated with Intermediate Physics for Medicine and Biology, Fig. 14.17.
The geometry associated with IPMB Fig. 14.17.
To help the reader learn more about the physics of OCT and Fig. 14.17, I have written two new homework problems. The solutions are included at the bottom of the post (upside down, to encourage readers to solve the problems themselves first). Enjoy!
Section 14.7

Problem 24 ⅓. This problem and the next explore the physics behind Fig. 14.17, which shows the velocity profile in a blood vessel measured using color Doppler optical coherence tomography. The data is based on an article by Yazdanfar et al. (2000). For this problem ignore the index of refraction of the tissue and assume θ = 60°.

(a) If the wavelength, λ, of the incident light is 832 nm and the wavelength bandwidth, Δλ, is 15 nm, determine the frequency, f, and the frequency bandwidth, Δf, in THz.

(b) The coherence time, τcoh, is equal to 1/(πΔf). Calculate τcoh in fs, and the coherence length, x2x1, in microns. The coherence length determines the spatial resolution of the measurement.

(c) Use Eq. 13.42 to derive an expression for the speed of blood flow in the direction of the light, v', in terms of the Doppler frequency shift, df. Assume that the speed of light, c, is much greater than v'. Calculate v' if df = 4 kHz. The Doppler technique measures the component of motion in the direction of the light. Determine the speed v along the vessel.

Problem 24 ⅔. The Doppler shift, df, of OCT data as a function of depth z across a blood vessel is given below. For viscous flow in a tube (Sec. 1.17), the blood speed varies parabolically across the vessel cross section (Eq. 1.37). Fit a parabola to this data of the form df = Az2 + Bz + C, and determine constants A, B, and C. Use these constants to find the peak value of df in this vessel, the location of the center of the vessel, and the vessel diameter (the width of the parabola when df = 0). The measured diameter corresponds to an oblique section at θ = 60°. Correct this result to get the true diameter.
   z (mm)    df (kHz)
0.15 3.26
0.20 5.20
0.25 6.12
0.30 6.02
0.35 4.89
0.40 2.75
I will give the final word to Yazdanfar, Rollins, and Izatt, who conclude
In summary, CDOCT has been applied for what is believed to be the first time to retinal blood-flow mapping in the human eye. Depth-resolved quantification of retinal hemodynamics may prove helpful in understanding the pathogenesis of several ocular diseases. Unlike fluorescein angiography, CDOCT is entirely noninvasive and does not require dilation of the pupil. Furthermore, CDOCT operates at longer wavelengths than does laser Doppler velocimetry, so light exposure times can be safely increased. CDOCT is believed to be the first technique for determining, with micrometer-scale resolution, the depth, diameter, and flow rate of blood vessels within the living retina.
Solution to new Homework Problem 24 ⅓
Solution to new Homework Problem 24 ⅓
Solution to new Homework Problem 24 ⅔.
Solution to new Homework Problem 24 ⅔.

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