Julien Chaput

The advent of full waveform methods in seismology has meant a rapid expansion imaging resolution and accuracy, and these methods have begun to show up in medical ultrasonic tomography. Unlike seismology however, frequencies in medical ultrasound have an upper range in the MHz, thus requiring typical sample rates in the 10s of MHz. Full waveform imaging requires the generation and recording of broadband waveforms with very high fidelity (i.e., very high end analog to digital converters, many receivers, extremely high data transfer rates in the 100s of GB/s). The high cost of such devices (~500k-1M$) thus lies in the datalogging system and not in the transducer/driver system at all. Conversely, the internet of things and DIY culture has spawned a slew of low-cost innovations in other medical fields (for instance, Experience of how to build an MRI machine from scratch - ScienceDirect), and we see no reason ultrasonic tomography can't be next.

We tackle in particular the main high-cost angle of building the device: The datalogging system, or, rather, the kind of data one needs to perform a high quality inversion. In practice, full waveform solutions are obtained via frequency stepping, that is, we can increase the stability of the inverse process can processing sequential sub-bands of the full waveform, thus starting from a smooth but globally attainable solution, and gradually adding less stable high frequency information into the mix. Given that the full spectrum of the waveform is redundant (i.e., you only need a subset of those frequencies to actually prevent cycle skipping in the resulting model), then we may choose instead to generate data one frequency at a time. Now why would we do this?

There are decades of signal processing innovations in, for example, the radar field, that allow for signal demodulation (i.e., the clever way that different radio stations for example don't interfere with each other) through removal of a carrier frequency. If a single frequency is present in the system and that frequency is known exactly, IQ demodulation results in a simple DC signal, which can be easily sampled by a very low cost ADC (i.e., averaged over 10k samples) and with a total data volume orders of magnitude lower than for a fully sampled time domain waveform. This of course comes at the cost of needing to record multiple sequential frequencies, but with multiple order of magnitude reduction in material costs! So a bit of math. Starting with the scalar wave equation, we assume separability:

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Substituting, we recover:

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And:

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This leads to the set of equations:

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With the first simply being the Helmholtz equation, and the second representing the time domain solution, satisfied by a harmonic time function. Thus, instead of simulating the entire time sequence for the wave equation, we opt to model the Helmholtz equation instead at a single frequency for a practically expected geometry, namely a 2D disk.

In practice, ultrasonic transducers produce pressure from a voltage input,​ and what is ultimately measured by the ADC is again voltage at the receiver set of transducers. For numerical convenience, a formal map is required for inversion, either Dirichlet to Neumann or vice versa, with full phase harmonic data being recovered by IQ (in phase and quadrature) demodulation to DC.

In order to test the invertibility of such a map for a dataset consisting of multiple harmonic frequencies and Fourier boundary forcing modes, we design a finite element simulation on a 2D disk featuring a realistic phantom with various embedded feature and sharp contrasts.
Given the high number of parameters in the system, the gradient descent step is handled by an adjoint formulation with both total variation and Tikhonov regularizations that sweep through high to low values as a function of sequential bandwidth steps. This numerical approach allows us to study the data requirements for a real-world device, but we anticipate the real inverse scheme to be handle by training the device itself on a series of realistic phantoms with known structures and approximate the forward/inverse process with a deep learning model, thus avoiding the necessary fine tuning of any numerical model to the real device.

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