A couple of weeks ago I talked about the feasibility of using nonuniform lattices sampled nonconcurrently to measure 3-dimensional vibrations in the cochlea. Without going back into too much detail, the heart of the issue is that the cochlea moves in three dimensions in time, and thus the function we are looking to measure is 4-dimensional.
Other groups have been able to sample quite roughly and map out some two-dimensional motion, as well as create some very rudimentary videos of this motion in space using nonconcurrent sampling. This week, with the help of a research scientist here at CUMC, I have been writing some software to measure this 4-dimensional function and visualize it in MATLAB. The results are very very pretty.
If you have no interest in the math involved in generating these figures and videos, then simply scroll down and enjoy. However, I feel it is of value to present why these videos serve as validation.
Our test setup is a small capillary with inner diameter 1.5 mm, which has been coated in a glue membrane about 1 micron thick. A small speaker was placed next to the capillary and the SDOCT system was set up to view it from above.
The sample monitor on the OCT shows us a top-down view of the membrane which we will use to get our bearings for the measurements we are about to take.
A B-Scan shows this membrane is truly there - the capillary is not hollow. It is about 700 nm - 1 micron thick, so it is smaller than the resolution of our system allows for. The motions we expect to see will be smaller than this thickness - in the 10s of nanometers. This is reasonable given our spectral domain phase microscopy method, which can sense sub-nanometer motion.
This is meant to validate our code, so we must know how this object should naturally behave to have a baseline to compare to. This is a solved problem - circular membrane vibration satisfies the two-dimensional wave equation with boundary conditions of no vibration at the boundary.
The solution to this equation is that all responses can be seen as a sum of modes in the membrane, not unlike the sine and cosine modes seen as the solution to the one-dimensional wave equation. Unfortunately, these modes are much more complicated - they have two parameters and depend on the Bessel Functions, which I see as relatively scary objects.
More specifically, there are frequencies at which certain very uniform patterns in membrane vibration occur, and these frequencies are determined by three constants - the speed of transverse waves in your membrane, the radius of the membrane and the roots of the Bessel functions of the first kind of all orders. The radius is known, and the roots of the Bessel functions for a two parameter family indexed by (order, root), where order takes all non-negative integer values and root takes all positive integer values. Many tables online give these roots, however the speed in the material is not known to us. It is dependent on some constants and thus we must conduct some tests to determine these modal frequencies.
The modes shown in the above figure tell us that as we increase the frequency, the center of the membrane will be a node at certain modes and an antinode at others. For mode (0,1), for example, the center will vibrate at a maximum when compared to other frequencies, while for mode (1,1), the center should not vibrate at all.
To determine our f1, and thereby all mode frequencies, we conduct a frequency sweep from 1 kHz to 40 kHz.
One can see that there are a number of maxima and minima spaced unevenly, and enveloped by an exponential sort of decay. The first maximum should correspond to the (0,1) mode, and we see this is at about 6 kHz. The first minimum should be the (1,1) mode, and so on. The shape of this curve does make some sense, as the heights of the maxima should fall off due to the decaying nature of the Bessel functions of the first kind.
So with this information, we can compute the speed of transverse waves in the membrane - it comes out to about 12 m/s, which seems vaguely correct. We can then start tackling the two-and three dimensional vibration problem with an expected output.
When we stimulate the cochlea in our experiments, we use a sound stimulus with many frequency components. In a linear system such as this one, a Fourier transform will allow us to separate the resultant response into its individual frequency components. If we hit the membrane, then, with a sound stimulus containing frequency components at a number of its modal frequencies, we can extract the individual modes as well as the total motion. If we see the shapes we expect for the given modes, it is likely working!
Our first attempt is at 2-D scanning, taken across the diameter of the membrane. We take 63 points in our scan, which takes about a minute. Extracting individual modes, we can graph the displacement as a function of space along our scan axis and time. For mode (1,1), for example, we get something like this:
The color, as well as the topological map drawn in the space-time plane helps us get our bearings. This is, no doubt, the (1,1) mode we are looking for. We can also look at a video on the scan axis, like this:
Higher order vibrations look a bit more interesting and it is often easier to gleam information on them from the top-down color map formed by looking at the surface plots from above. For example, here is mode (1,2):
By summing all 40 frequency components, we get the complicated true vibration of our membrane in time:
This all looks to show that high-resolution 2-D vibrometry can be performed relatively quickly without any tricks being pulled.
Our motion to 3 dimensions required the choice of a lattice. Our first choice is the simple rectangular lattice, and we use an 11 point by 11 point grid across a 1mm x 1mm area of the membrane.
The surface plots are now what must be animated to give an understanding of motion. Here is the (1,1) mode for example:
The higher order modes start to show some quantization issues from our low spatial sampling, but are still easily made out. Here is mode (0,3) for example:
A sum of all frequency components follows.
While not all modes are spatially contained enough for us to get a perfect reconstruction with this lattice, higher frequency components have significantly lower amplitudes, and thus spatial aliasing has a relatively small effect on the resulting videos.
These videos, unfortunately, take about 20 minutes to take. A less dense lattice would be necessary for recovery studies, but 20 minutes is likely fine for a healthy cochlea.
I wanted to share these lovely videos not just because they are pretty, but also to show the strength of OCT vibrometry. Using phase methods, we are able to see sub-nanometer vibrations which near-directly match our analytic results. It also shows an interesting, unexpected use of the Bessel functions in system validation!
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