Photoacoustic Tomography

In March 2018 Gudrun had a day available in London when
travelling back from the FENICS workshop in Oxford. She contacted
a few people working in mathematics at the University College
London (ULC) and asked for their time in order to talk about
their research. In the end she brought back three episodes for
the podcast. This is the second of these conversations.


Gudrun talks to Marta Betcke. Marta is associate professor at the
UCL Department of Computer Science, member of Centre for Inverse
Problems and Centre for Medical Image Computing. She has been in
London since 2009. Before that she was a postdoc in the
Department of Mathematics at the University of Manchester working
on novel X-ray CT scanners for airport baggage screening.


This was her entrance into Photoacoustic tomography (PAT), the
topic Gudrun and Marta talk about at length in the episode. PAT
is a way to see inside objects without destroying them. It makes
images of body interiors. There the contrast is due to optical
absorption, while the information is carried to the surface of
the tissue by ultrasound. This is like measuring the sound of
thunder after lightning. Measurements together with mathematics
provide ideas about the inside. The technique combines the best
of light and sound since good contrast from optical part - though
with low resolution - while ultrasound has good resolution but
poor contrast (since not enough absorption is going on).


In PAT, the measurements are recorded at the surface of the
tissue by an array of ultrasound sensors. Each of that only
detects the field over a small volume of space, and the
measurement continues only for a finite time. In order to form a
PAT image, it is necessary to solve an inverse initial value
problem by inferring an initial acoustic pressure distribution
from measured acoustic time series. In many practical imaging
scenarios it is not possible to obtain the full data, or the data
may be sub-sampled for faster data acquisition. Then numerical
models of wave propagation can be used within the variational
image reconstruction framework to find a regularized
least-squares solution of an optimization problem.


Assuming homogeneous acoustic properties and the absence of
acoustic absorption the measured time series can be related to
the initial pressure distribution via the spherical mean Radon
transform. Integral geometry can be used to derive direct,
explicit inversion formulae for certain sensor geometries, such
as e.g. spherical arrays.


At the moment PAT is predominantly used in preclinical setting,
to image tomours and vasculature in small animals. Breast
imaging, endoscopic fetus imaging as well as monitoring of
perfusion and drug metabolism are subject of intensive ongoing
research.


The forward problem is related to the absorption of the light and
modeled by the wave equation assuming instanteneous absorption
and the resulting thearmal expansion. In our case, an optical
ultrasound sensor records acoustic waves over time, i.e.
providing time series with desired spacial and temporal
resolution. Given complete data, then one can mathematically
reverse the time direction and find out the original object.


Often it is not possible to collect a complete data due to e.g.
single sided access to the object as in breast imaging or
underlying dynamics happening on a faster rate than one can
collect data. In such situations one can formulate the problem in
variational framework using regularisation to compensate for the
missing data.


In particular in subsampling scenario, one would like to use
raytracing methods as they scale linearly with the number of
sensors. Marta's group is developing flexible acoustic solvers
based on ray tracing discretisation of the Green's formulas. They
cannot handle reflections but it is approximately correct to
assume this to be true as the soundspeed variation is soft tissue
is subtle. These solvers can be deployed alongside with
stochastic iterative solvers for efficient solution of the
variational formulation.


Marta went to school in Poland. She finished her education there
in a very selected school and loved math due to a great math
teacher (which was also her aunt). She decidede to study Computer
Sciences, since there she saw more chances on the job market.
When moving to Germany her degree was not accepted, so she had to
enrol again. This time for Computer Sciences and Engineering at
the Hamburg University of Technology. After that she worked on
her PhD in the small group of Heinrich Voss there. She had good
computing skills and fit in very well. When she finished there
she was married and had to solve a two body problem, which
brought the couple to Manchester, where a double position was
offered.
Now both have a permanent position in London.



References

M. Betcke e.a.: Model-Based Learning for Accelerated,
Limited-View 3-D Photoacoustic Tomography IEEE Transactions on
Medical Imaging 37, 1382 - 1393, 2018.

F. Rullan & M. Betcke: Hamilton-Green solver for the
forward and adjoint problems in photoacoustic tomography archive,
2018.

M. Betcke e.a.: On the adjoint operator in photoacoustic
tomography Inverse Problems 32, 115012, 2016. doi

C. Lutzweiler and D. Razansky: Optoacoustic imaging and
tomography - reconstruction approaches and outstanding challenges
in image performance and quantification, Sensors 13 7345, 2013.
doi: 10.3390/s130607345




Podcasts

G. Thäter, K. Page: Embryonic Patterns, Gespräch im
Modellansatz Podcast, Folge 161, Fakultät für Mathematik,
Karlsruher Institut für Technologie (KIT), 2018.



F. Cakoni, G. Thäter: Linear Sampling, Conversation im
Modellansatz Podcast, Episode 226, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2019.

G. Thäter, R. Aceska: Dynamic Sampling, Gespräch im
Modellansatz Podcast, Folge 173, Fakultät für Mathematik,
Karlsruher Institut für Technologie (KIT), 2018.

S. Fliss, G. Thäter: Transparent Boundaries. Conversation in
the Modellansatz Podcast episode 75, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.

S. Hollborn: Impedanztomographie. Gespräch mit G. Thäter im
Modellansatz Podcast, Folge 68, Fakultät für Mathematik,
Karlsruher Institut für Technologie (KIT), 2015.

M. Kray, G. Thäter: Splitting Waves. Conversation in the
Modellansatz Podcast episode 62, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.

F. Sayas, G. Thäter: Acoustic scattering. Conversation in the
Modellansatz Podcast episode 58, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.