Numerical rate function determination in partial differential equations modeling cell population dynamics.

This paper introduces a method to solve the inverse problem of determining an unknown rate function in a partial differential equation (PDE) based on discrete measurements of the modeled quantity. The focus is put on a size-structured population balance equation (PBE) predicting the evolution of the number distribution of a single cell population as a function of the size variable. Since the inverse problem at hand is ill-posed, an adequate regularization scheme is required to avoid amplification of measurement errors in the solution method. The technique developed in this work to determine a rate function in a PBE is based on the approximate inverse method, a pointwise regularization scheme, which employs two key ideas. Firstly, the mollification in the directions of time and size variables are separated. Secondly, instable numerical data derivatives are circumvented by shifting the differentiation to an analytically given function. To examine the performance of the introduced scheme, adapted test scenarios have been designed with different levels of data disturbance simulating the model and measurement errors in practice. The success of the method is substantiated by visualizing the results of these numerical experiments.

Principal Component Analysis of gait in Parkinson's disease: relevance of gait velocity.

Principal Component Analysis (PCA) is a method to estimate the relation between data points. We used PCA to analyse movements of the upper and lower extremities during treadmill walking in healthy subjects and two groups of Parkinsonian patients. Healthy subjects (n=35) showed a typical pattern with high values of PC1 and low values in a descending order of PC2-PC4. Increase of speed resulted in a significant increase of PC1 and a significant decrease of the following PC's. In more severely affected patients (n=19, UPDRS>20), PC1 was significantly decreased and PC2-PC4 were significantly increased compared to healthy subjects. Speed could be increased only within a small range without corresponding changes of the PC's. In less severely affected patients (n=17), significant differences of the PC's were only found with fast pace. Separate analysis of arms and legs revealed that these changes are only due to altered movements of the arm. Analysis of the pattern of PC's in response to changes of gait velocities reveal alterations even in less severely affected Parkinsonian patients. The changes of the PC's with higher gait velocities in healthy subjects are suggestive of an increase of intersegmental coordination. This is impaired even in less severely affected Parkinsonian patients.

Stochastic modelling of biased cell migration and collagen matrix modification.

Matrix dynamics plays a crucial role in several physiological and pathological processes. In this paper we develop a model framework, which describes the temporal fibre network evolution depending on the influence of migrating fibroblasts. The cells are regarded as discrete objects in the plane, whose velocities are determined by a generalised Langevin equation. For its solution we verify existence and uniqueness. The courses of the trajectories are affected by two external impulses, chemotaxis and contact guidance, respectively. The extracellular matrix is described by a continuous vector field which contains both information on density and orientation of the fibrous material. Modelling dynamic interaction between the discrete and the continuum variables is an essential point of this paper. In particular, the smoothing of the fluctuating paths plays a key role. Besides a detailed description of the formulated equations, we also supply the condensed pseudo code of the algorithm. We investigate several examples and present results both from artificial and real data.

Computing reconstruction kernels for circular 3-D cone beam tomography.

In this paper, we present techniques for deriving inversion algorithms in 3-D computer tomography. To this end, we introduce the mathematical model and apply a general strategy, the so-called approximate inverse, for deriving both exact and numerical inversion formulas. Using further approximations, we derive a 2-D shift-invariant filter for circular-orbit cone-beam imaging. Results from real data are presented.