Low-dose medical X-ray imaging

Research report by Professor Samuli Siltanen

Note: this page is a bit old and not anymore updated. For a more recent explanation of tomographic X-ray imaging please go to this page and click "X-ray Tomography."

Contents:
Introduction
First test case: single tooth and total variation prior
Second test case: intraoral imaging, head phantom and total variation prior
Highlight: 3D imaging using a panoramic device
Modified level set method for X-ray tomography


Introduction

This is a review of a large body of research work on medical X-ray imaging
done since the year 2001, resulting in more than a dozen scientific
articles, five patents, and one commercial product (www.vt-cube.com).
A large number of people were involved in the research effort; their names
appear below in the references to publications.

The story started around the year 2000 when I was working as a research
scientist at Instrumentarium Imaging, a company at that time specialized in
designing and manufacturing medical X-ray imaging devices. Here is a picture
of some of the products:

From left to right: dental panoramic imaging device, mammography device,
and C-arm surgical imaging device.

The starting point of the research was the desire of using the above X-ray
machines to collect projection data from several directions and to compute
a three-dimensional (3D) reconstruction of tissue. This is different from
traditional computerized tomography (CT), where one collects data from
more than a hundred directions from full angle of view, enabling very
accurate 3D reconstructions of tissue. The data collected with the above
devices has typically a limited angle of view and consists only of at most
a dozen images.

The philosophy of the proposed 3D imaging is the following: Let the doctor
collect whatever data is practical to measure with the equipment at hand.
Our task is to create mathematical methods that compute 3D reconstructions
with high enough quality for the clinical task in question.

A research project was started in 2001, funded by Finnish Technology Agency
(TEKES), involving researchers from both industry and academia, with the goal
of developing new low-dose, sparse-data 3D reconstruction algorithms.

We found quite soon that traditional filtered back-projection (FBP) methods
are not optimal for this kind of imaging, since the incomplete data violates
the assumptions of FBP. We proceeded to apply Bayesian inversion, a flexible
and noise-robust family of algorithms. It turned out that Bayesian inversion
is very well-suited for this application. See the following article as well as
the examples below. Siltanen, Kolehmainen, Järvenpää, Kaipio, Koistinen,
Lassas, Pirttilä and Somersalo
, Statistical inversion for medical X-ray tomo-
graphy with few radiographs I: General theory
, Physics in Medicine and
Biology 48 (2003), pp. 1437-1463. PDF (407 KB)

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First test case: single tooth and total variation prior

We decided to start our study of Bayesian 3D X-ray imaging with a very simple
situation. A single wisdom tooth, generously donated to science by my friend
Helena Sarlin, was attached to a rotating platform and imaged from full angle
of view. Here is a picture of our laboratory setup:

Let's take a closer look of the tooth, the rotating platform, and the intraoral
digital X-ray detector attached to the wall:

We record 23 projection images as shown in the drawing above. Each image
has 664 x 872 pixels; below are some examples of how the projection images
look like. Note that our data set cover full angle of view because we want to
compute full-angle reconstruction for comparison with the limited angle
reconstruction done using only the angles marked by white discs above.

Here are two reconstructed horizontal slices through the tooth. Left slice is
computed using full-angle data, and the right slice is computed using the 9
projections from roughly 65 degree angle of view. Note the typical elongation
of the limited angle reconstruction towards the direction of the measured rays.
We compute a stack of 400 such two-dimensional reconstructions, forming a
three-dimensional reconstruction of the whole tooth.

According to Todd Quinto's seminal 1993 article on stable singularities in
tomography,we expect to see certain crisp boundaries in the horizontal slices.
Roughly speaking, they are the jump discontinuities tangented by some of the
X-rays in the data set; this can be seen in the above two slices by comparing
the boundaries. We make use of this fact by stacking all the 400 horizontal
reconstructed slices on top of each other and slicing the stack vertically. Here
is an example of such a slice, again computed from both full and limited data:

As you can see, the limited-angle result is not so different from the full-angle
result when sliced this way. On the other hand, a vertical slice perpendicular
to the above, computed from the limited angle data, would be smooth and very
far from the truth. This is an example of a Bayesian limited angle reconstruction
not being perfect but still showing some aspects of tissue reliably.

The above computations are based on the use of (approximate) total variation
prior and the computation of the Maximum a Posteriori (MAP) estimate. For
details of computation and more images of reconstructions, see the following
article:

Kolehmainen V, Siltanen S, Jarvenpaa S, Kaipio J P, Koistinen P, Lassas M,
Pirttila J and Somersalo E
, Statistical inversion for medical X-ray tomography
with few radiographs II: Application to dental radiology
,
Physics in Medicine and Biology 48 (2003), pp. 1465-1490. PDF (1.2 MB)

Ville Kolehmainen and me in year 2001 after measuring for 11 hours in a row:

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Second test case: intraoral imaging, head phantom
and total variation prior

Encouraged by the above reconstructions of a single tooth, we decided to try a
more realistic measurement modelling the chairside intraoral X-ray imaging
modality used routinely in dental clinics. The idea is to insert a small digital
X-ray detector inside the mouth of a patient, let the patient hold the detector
against her teeth, and take for example five images from different directions.
Using a small steel ball attached to the crown of one of the teeth enables the
calibration of the projection directions.

We used a so-called head phantom in our measurement. It is an actual human
skull covered by rubber. The yellow device on the left bottom corner in the
image below is the X-ray source, which was moved into different positions for
recording projection data. As a side remark, the upside down positioning of the
head phantom is not the suggested measurement configuration; our tripod was
broken and could only hold the phantom steady in the way shown.

We collected seven radiographs from a 60 degree angle of view according to
the drawing below on the left. On the right you see one of the images, each
having 664 x 872 pixels. Consequently our data consists of roughly 4 million
pixel values. We divided the target tissue into some 42 million voxels, so our
data matrix has size 4000000x42000000. It is not advisable to construct such
a large matrix explicitly, so we relied on matrix-free iterative reconstruction
methods.

Here are two vertical slices through the teeth. Note that on the left we see only
one of the three roots of the tooth in the middle, whereas on the right we see the
two other roots. This shows that our method can separate features at different
depths inside the tissue. This is crucial for example in the detection of alveolar
disease, or bone loss between teeth. The loss may not be visible in a single
X-ray image, but a slice at the correct depth will reveal the condition.

The above computations are based on the use of (approximate) total variation
prior and the computation of the Maximum a Posteriori (MAP) estimate. For
details of computation and more images of reconstructions, see the following
articles:

Kolehmainen V, Siltanen S, Jarvenpaa S, Kaipio J P, Koistinen P, Lassas M,
Pirttila J and Somersalo E
, Statistical inversion for medical X-ray tomography
with few radiographs II: Application to dental radiology
,
Physics in Medicine and Biology 48 (2003), pp. 1465-1490. PDF (1.2 MB)

Kolehmainen V, Vanne A, Siltanen S, Jarvenpaa S, Kaipio J P, Lassas M
and Kalke M
, Bayesian Inversion Method for 3-D Dental X-ray Imaging.
Elektrotechnik & Informationstechnik 124 (2007), pp. 248-253. PDF (179 KB).

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Highlight: 3D imaging using a panoramic device

The 3D X-ray imaging project has so far resulted in one commercial product,
the VT tomograph (www.vt-cube.com). The idea is to use a panoramic dental
imaging device in a novel way to produce 3D information helpful for dental
implant planning.

Let me first explain the idea of panoramic imaging. In the 1950's there was a
competition among dental radiologists to invent a technique for imaging all
the teeth and nearby structures simultaneously. The race was won by a Finnish
researcher, Professor Yrjö Paatero, who was at the time working at University
of Washington in Seattle, USA. Paatero's invention was based on an ingenious
geometric movement of the X-ray source and the film holder, resulting in one
X-ray image showing a curved sharp layer inside the teeth. Here is a schematic
illustration of Paatero's geometric principle:

The black dot in the middle is the center of rotation. The X-ray source (black
disc on the left) and the detector plane (on the right) are rigidly attached to the
center of rotation. During the rotation, the detector moves linearly along the
detector plane. (With CCD detectors this movement is realized by translating
the pixels' potential wells electronically.) Now pay attention to the image of a
detail (red square) located away from the center of rotation. When the X-rays
are horizontal, the detail is mapped in the middle of the detector. Later the rays
travel at an angle (dashed line), and the detail is mapped to the edge of the
detector. If the pixels are moved with exactly the speed of the detail's image
traveling on the detector plane, a sharp layer along the red circle is recorded.
Any details located away from the red circle are blurred. When the sharp layer
is carefully aligned with the dental arc, the result is a panoramic X-ray image:

Yrjö Paatero returned to Finland in the 1960's and started a company called
Palomex manufacturing panoramic imaging devices. Since then the panoramic
image has been a basic workhorse of dental clinics.

There are some situations, however, when the information contained in a 2D
X-ray image, panoramic or otherwise, is not enough for the clinical task. For
example, in dental implant planning the aim is to replace a missing tooth by an
artificial one which is screwed into bone.

The important thing is to measure the distance to the nerve located inside the
mandible. The hole should be drilled deep enough for the implant to hold, but
not so deep that the nerve is damaged as the result can be facial paralysis. This
measurement cannot be reliably made from a projection image or a panoramic
image because of geometric distortions.

We reprogrammed the panoramic device so that it collects limited-angle data
as shown in the drawing below. Using our new 3D reconstruction methods, we
can compute a two dimensional slice enabling a reliable measurement.

The nice thing about the above 3D imaging method is that a dental clinic can
use the existing equipment: a software update turns a 2D panoramic device
into a 3D tomographic instrument. See www.vt-cube.com for more information.

Furthermore, we explored the possibility of complementing the projection data
by an additional panoramic image. We found out that the reconstruction quality
can be improved somewhat by using the extra data.

Hyvonen N, Kalke M, Lassas M, Setala H and Siltanen S, Three-dimensional
X-ray imaging using hybrid data collected with a digital panoramic device.

Inverse Problems and Imaging 4(2), pp. 257-271. PDF (724 KB)

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Modified level set method for X-ray tomography

Reconstructions from limited angle data are often stretched along the directions
of the X-rays, almost regardless of the inversion algorithm. Removing those
stretching artifacts has been one of the central goals in the research project.
The most successful approach so far is the modified level set method.

The core object in every level set method is the so-called level set function,
whose zero level set defines the boundary between two different components.
In inverse problems, the unknown object to be reconstructed is represented
differently (often as constants) inside and outside the domain defined by the
level set. The level set function deforms according to a certain time evolution
scheme, resulting in flexible and topologically variable domains determined
by the zero level set.

While implementing the classical piecewise constant level set method for our
tomographic application we noticed that the level set function itself looks like
a reconstruction of the X-ray attenuation coefficient. We decided to represent
the attenuation coefficient by zero outside the level set (providing natural
enforcement of non-negativity) and by the level set function itself inside the
level set. The result is an inversion method with very efficient truncation of the
reconstruction to zero.

Here is an example using the synthetic Shepp-Logan phantom. We collected
10 projection images with full angle of view and reconstructed the phatntom
(left) using filtered back-projection (middle) and our level set method (right).

We repeated the above experiment with real data measured using a surgical
C-arm device from a knee specimen. Left: full-angle reconstruction from 60
projections using the level set method, middle: filtered back-projection recon-
struction from 10 projections, right: level set reconstruction from 10 projections.

In the case of limited angle tomography, our novel level set method turns out to
be quite effective in avoiding stretching artifacts. Left: level set reconstruction
from full-angle data, middle: back-projection (without filtering) from limited-
angle data, right: level set reconstruction from limited-angle data.

Using the level set function itself to represent the attenuation coefficient inside
the level set has an additional advantage. Namely, we can give a convergence
proof for our method. See the article Kolehmainen V, Lassas M and Siltanen S,
Limited data X-ray tomography using nonlinear evolution equations,
SIAM Journal of Scientific Computation 30 (2008), pp. 14131429. PDF (642 KB)

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Research home page of Samuli Siltanen