Biomedical Engineering Reference
In-Depth Information
dimensions, a spectral dimension shows up additionally. The corresponding
mathematical model is the Radon transform in four dimensions. This tech-
nology is presently studied in the stages of pharmaceutical research and animal
experiments. However, due to the limitations in field strength, the data in Radon
space cannot be sampled completely with the consequence that a limited angle
problem has to be solved. Theoretically, the desired distribution would be
uniquely determined, if all data in the restricted range were available, but
instabilities and strong artifacts complicate the reconstruction problem.
3. Ultrasound CT: In this approach, the sender and receiver are spatially sepa-
rated, the corresponding mathematical model is an inverse scattering problem
for the determination of the spatially varying sound impedance and the scat-
tering properties. The difficulty here is that, in contrast to CT, the paths of the
waves depend upon the variable to be computed, which makes the problem
highly nonlinear. A linearization of the problem via Born or Rytov approxi-
mation neglects the effects of multiple scattering, and is therefore not suffi-
ciently accurate. That is why ultrasound tomography today is still a challenge to
mathematics and algorithm development.
4. Transmission electron microscopy (TEM): For the visualization of biomole-
cules by TEM, various approaches are pursued. If one does not aim at aver-
aging over many probes of the same kind, again a limited angle problem arises.
In addition, wave phenomena enter for small-sized objects, leading, as in
ultrasound tomography, to nonlinear inverse scattering problems. Fortunately,
linearization is feasible here, which facilitates the development of algorithms
significantly.
5. Phase-contrast tomography: In this technology, where complex-valued sizes
have to be reconstructed, linearizations are also applicable. The phase supplies
information even when the density differences within the object are extremely
small. Due to different scanning geometries, medical application still generates
challenging problems for the development of algorithms.
6. Diffusion tensor MRI: This method provides a tensor at each reconstruction
point, thus bearing information concerning the diffusivity of water molecules in
tissue. In this, reconstruction and regularization are performed separately. Only
after having computed tensors, properties of the tensor like symmetry or
positive definiteness are produced point by point.
The above list of imaging techniques under present development is by no means
complete. Methods like impedance tomography are studied as well as those
applying light to detect objects close to the skin. In all of the mentioned measuring
techniques, the technological development is so advanced that the solution of the
associated mathematical problems like modeling, determination of achievable
resolution, and development of efficient algorithms will yield a considerable
innovation thrust.
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