[Mathematical modelling in systems biology. Simulation of the desmoplastic stromal reaction as an example]. / Mathematische Modellierung in der Systembiologie. Beispiel einer Simulation der desmoplastischen Stromareaktion.

A mathematical model of collagen fiber mesh formation was created to evaluate the possible role of chemotaxis and haptotaxis in the histomorphology of a desmoplastic stromal reaction (DSR). Fibroblasts were mathematicaly interpreted as mobile discrete objects, characterized by their velocity and position, both dependent on time. This resulted in cell migration paths, commonly termed "trajectories" which are modulated as stochastic process. The implementation of chemotactic effects requires knowledge of the concentration and distribution of the appropriate chemical substance in the scenario. A simplistic model assumption allows the calculation of a numerical solution of the resulting diffusion equation. Adding haptotaxis necessitates the simulation of the extracellular matrix (ECM). The fiber distribution is modeled as a vector field which contains information on both, fiber density and direction. The production of new fibers is based on ordinary differential equations coupled with the migratory behavior of the cells. Filters help smooth the trajectories. Appropriate visualization allows a direct comparison of the simulation results with histomorphology. Matches between computed data and their real counterparts indicate that the development of mathematical models is appropriate to describe and forecast the course of DSR. This makes systems biology a stepping stone to improving biomedical research.

A reconstruction approach for imaging in 3D cone beam vector field tomography.

3D cone beam vector field tomography (VFT) aims for reconstructing and visualizing the velocity field of a moving fluid by measuring line integrals of projections of the vector field. The data are obtained by ultrasound measurements along a scanning curve which surrounds the object. From a mathematical point of view, we have to deal with the inversion of the vectorial cone beam transform. Since the vectorial cone beam transform of any gradient vector field with compact support is identically equal to zero, we can only hope to reconstruct the solenoidal part of an arbitrary vector field. In this paper we will at first summarize important properties of the cone beam transform for three-dimensional solenoidal vector fields and then propose a solution approach based on the method of approximate inverse. In this context, we intensively make use of results from scalar 3D computerized tomography. The findings presented in the paper will continuously be illustrated by pictures from first numerical experiments done with exact, simulated data.

Complete sets of radiating and nonradiating parts of a source and their fields with applications in inverse scattering limited-angle problems.

Many algorithms applied in inverse scattering problems use source-field systems instead of the direct computation of the unknown scatterer. It is well known that the resulting source problem does not have a unique solution, since certain parts of the source totally vanish outside of the reconstruction area. This paper provides for the two-dimensional case special sets of functions, which include all radiating and all nonradiating parts of the source. These sets are used to solve an acoustic inverse problem in two steps. The problem under discussion consists of determining an inhomogeneous obstacle supported in a part of a disc, from data, known for a subset of a two-dimensional circle. In a first step, the radiating parts are computed by solving a linear problem. The second step is nonlinear and consists of determining the nonradiating parts.

Numerical aspects of spatio-temporal current density reconstruction from EEG-/MEG-data.

The determination of the sources of electric activity inside the brain from electric and magnetic measurements on the surface of the head is known to be an ill-posed problem. In this paper, a new algorithm which takes temporal a priori information modeled by the smooth activation model into account is described and compared with existing algorithms such as Tikhonov-Phillips.

Spatio-temporal current density reconstruction (stCDR) from EEG/MEG-data.

Among the different approaches to the bioelectromagnetic inverse problem, the current-density reconstruction methods (CDR) provide the most general solutions. Since the inverse problem does not have a unique solution, model assumptions have to be taken into account. Multi-channel measurements contain not only spatial, but also temporal information about the sources, so a naturally extension to existing methods leads to spatio-temporal model constraints. Spatio-temporal CDR's (stCDR) have been tested in simplified volume conductor models, assuming different spatial model constraints and a smooth temporal activation model. Comparison to existing spatial model constraints showed a significant improvement of spatial and temporal resolution of the reconstructed sources for the spatio-temporal models especial in noisy data.

Contour reconstruction in 3-D X-ray CT.

The authors derives an algorithm for reconstructing contours of an object from 3D cone beam X-ray data. By reducing the amount of the searched-for information, contours, or density jumps instead of the densities themselves, the authors are able to develop fast algorithms for data incomplete with respect to both the movement of the X-ray source and the detector reading. The method is related to local or Lambda tomography. Numerical simulations show the efficiency of the algorithm.