Alex Barnett: Nontechnical introduction to my research

(updated Dec 2004)

My work centers around two main issues:
  1. the nature of resonant modes occuring in wave and quantum systems
  2. 'seeing' inside the human body (without surgery, that is, noninvasively) by modelling how light diffuses through tissue.
A lot of what I do involves getting computers to calculate things (equations) that approximate real-world phenomena. Here is an introduction to some of the motivation, methods, and questions.

Quantum chaos

Imagine a drum stretched across a frame of a certain shape. When you hit the drum, different frequencies are excited, each with its own `mode' of vibration. (They rapidly decay, giving the drum its characteristic sound). For the lowest frequency mode, the whole membrane will move together (in phase), and higher modes will have increasingly complicated patterns, involving oscillations with decreasing wavelength. For example...
Lowest 10 modes of of a drum shape formed from two straight lines and two arcs of circles. Red indicates upward (positive) motion at some instant of time, and blue downward (negative). (click to enlarge picture)
What happens as you go to much higher modes---higher in the `spectrum' of frequencies? Different drum shapes give different behaviours, depending on whether the shape is `chaotic' or not. Here `chaos' refers to a related problem, that of a point particle (imagine a billiard ball) bouncing around a `billiard table' of that shape:
Motion of a billiard ball bouncing inside regular and chaotic tables (click to enlarge)
The motion (dynamics) depends on the shape, and can be regular, chaotic, or some mixture (there are also some more exotic categories). There is controversy as to how much the billiard problem (the `classical' problem) influences the drum problem (the `quantum' problem). The equation for drum modes (the Laplace eigenvalue equation) is the same as the quantum-mechanics of a particle trapped in a cavity, hence the term `quantum', and the strong connection to physics. In fact, the drum problem emerged from classical physics (acoustics and electro-magnetism), and already in 1917 Einstein was concerned about the modes for chaotic systems The field of quantum chaos has blossomed since the 1980's. Questions I study include: In the limit of high mode number, do the modes become uniformly distributed across the shape, or do some modes cluster ('scar') along orbits where the billiard ball repeats its motion? What are the correlations between mode patterns? What are the statistics of `matrix elements' (overlaps with each other) of these modes? These statistics have real-world consequences for the heating (dissipation) rate in driven quantum systems, such as electrons trapped in tiny, cold micro-chip layers called `quantum dots'. Many of these issues have emerged from my doctoral work with Rick Heller's group at Harvard Physics Department.

Such questions require semiclassics (the theory of how waves behave in the limit of short wavelengths), and also large-scale numerical calculation of the modes. I have focussed on devising new and very efficient algorithms that allow a computer to be used to calculate these modes for a large variety of drum shapes, including with corners. This requires analysis of the properties of the modes themselves. As the wavelength gets shorter, the problem gets more challenging, however my methods become relatively more efficient compared to any other known methods. I typically compute many thousands of modes at around the 100,000th mode. Here the efficiency increase is about a thousand times! These methods have technological applications in designing micro-cavity lasers, less than a hundredth of a millimeter in size, for fiber-optic communication, and more generally in a wealth of acoustic and waveguide problems. I continue to collaborate with physicists, applied and semiclassical mathematicians, and even number theorists (who care about quantum ergodicity too!).

Medical imaging

There has been an explosion of research on how the human brain functions in the last 15 years, driven largely by functional Magnetic Resonance Imaging (fMRI) technology via the discovery that when a part of the brain is active, it `blushes', that is, has increased blood flow and oxygen level. However, it is also possible to map blood oxygen levels in the cortex (which lies mainly in the first centimeter of depth from the inside of the skull) using near-infrared light, with sources and detectors placed in contact with the scalp. Shine a flashlight through the tissue of your fingers and you will see only red light gets through, and that the light diffuses (otherwise humans would be transparent!)
The basic idea of Diffuse Optical Tomographic measurements (click to enlarge)
Extracting useful information from the detected light signals is called Diffuse Optical Tomography (DOT). As you might imagine, this method is much cheaper and simpler than fMRI, but also allows rapid imaging (hundreds of `frames' per second) and can be sensitive to different chemicals (eg deoxy-hemaglobin and oxy-hemaglobin).
DOT is so versatile that functional brain imaging can be performed even on a baby who is moving around and reacting to stimuli
However, the mathematics involved in some sense is harder, because no direct 'image' is produced, Rather we have to solve for (that is, adjust in a computer simulation) some model of what optically is inside the head until we get similar detected signals. This is called a nonlinear inverse problem, and since there are many nearly equally-correct answers, the problem is called `ill-posed'. The image we are looking for is maps of absorption and scattering strength of the tissues, at the various wavelengths of light used (typically 700-900 nm).

I have worked on fast numerical methods to calculate how short pulses of light diffuse through the tissue of the human head, in the complicated geometry of the scalp, skull and brain. However, there is much to be improved in these methods. Speed is important since solving the inverse problem requires using this `forward' simulation many times. Since the measurements (detected light intensities as a function of time, on the scale of a billionth of a second) are noisy, statistical methods are appropriate. I favour Bayesian methods, which tell you exactly what you have learned about your image (parameters in your model), and no more, in the sense of a probability distribution. The price you pay is that these methods are quite slow. Questions remain such as: what is the ultimate practical resolution of DOT in the human brain? What is the uncertainty in measured parameters of regions of the brain, given a certain noise? How much uncertainty can there be in calibration parameters of the experiment? What are optimal patterns of sources and detectors on the head surface?

I collaborate with David Boas's Photon Migration Imaging lab at the Martinos Center (Radiology Department) at Massachusetts General Hospital and Harvard Medical School, where many DOT techniques are being developed.


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