Curvature and Torsion of Particle Paths 

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Hydrodynamic Flow through Pore Space

The transport of cells through a porous phase and to the activated surface is a convec-tive process. A hydrodynamic flow path equipped with torsion allows to hit the concave surfaces for the floating cells. The convective flow through the pore space induces the boundary conditions for the selective initial adhesion of target cells to the activated surface. For an initial adhesion the cells have got in direct contact with the activated surface. The floating cell might drift out a zone of high volume flow velocity into a vol-ume element with low volume flow nearby the activated surface. Thereby, it might be attracted into geometric contact by mechanic impact of cell mass, by micro-turbulent flow and/or electrostatic attraction. The cells in geometric contact might have an av-erage residence time at the selective surface which is required to establish the specific molecular interaction between the binder molecules of the activated surface and the so-called reporter groups which are exposed at the surface of the target cells. Considering the complex pattern of convective transport of cells and the multistep binding process, it is very difficult to predict the conditions for most selective or most productive binding of target cells. Thus, the design of experiments approach is required for the development of an efficient cell separation device. The prerequisite for a successful design of experiment approach is the uniformity of microscopic process conditions in all volume elements of the stationary phase which are flown through by the suspended target cells. As described above, the most suitable geometry of the stationary phase is a defined isotropic structure as a multiplication of 3D unit elements, e.g. a face removed Weaire-Phelan foam (see Fig. 1.2) or an open foam with a narrow random distribution of pore diameters. The macroscopic flow of cells through the stationary phase has to be invariant for every 3D unit element. The flow can be generated either by moving a fluid medium through a closed device which contains the stationary phase or by moving the stationary phase in an open device through a fluid medium. In both cases the flow conditions of the entrance of the stationary phase have to be very similar to the flow conditions during percolation and at the outlet. This setup cannot be realized by a chromatographic column which has an inlet and an outlet fluid connection with diameters far smaller than that of the column bed or stationary phase. Large scale chromatography columns are often equipped with flow distributing spare parts at the entrance. These functional elements are efficient for solutions of molecules in which diffusion controls the mass transfer. In contrast, the movement of suspended cells floating through a stationary phase is convective. An entrance flow which passes narrow inlets or even flow distributors results in an irregular movement of cells characterized by jet streams of high cell density and volumes in be-tween with low cell densities. This uneven and not controllable flow scheme renders any tuning of the cell binding process impossible. As a simple alternative, a set of parallel capillaries can be used to generate an even and controllable flow. But capillary struc-tures have typically very low property of cells to surface contact event and therefore a low binding capacity for cells.

Investigation of Chromatographic Filters by Tomogra-phy

The classical investigation of chromatographic filters is focused on the cell adhesion realized by the chemical activation of the surface and the composition of the mobile phase. The investigation setup is normally simplified to one-dimensional flow over thin-layered stationary phase, which can be easily observed by microscopes [Oh et al., 2015]. However, in this thesis, the influence of the geometry of the pore space and the 3D flow through it is the matter of investigation. Therefore, microscopes are no longer possible and the more sufficient method of computed tomography is used for the investigation.
Tomography is the process to create 3D images by using projections of a rotating 3D object. When discrete calculations are mandatory for the tomography process, it is called computed tomography (CT). In this work, the 3D images have at least a resolution more accurate than 10 ➭m. So, to distinguish the data under interest from CT-scans in medical application with a resolution of millimeter, the process is specified as ➭CT.
Tomography can be classified by the shape of the beam. There are ➭CT with cone beam and collimated beam. Cone beam ➭CT is used in laboratory CT scanners with a lens less magnification. The magnification factor depends only on the distance between the sample and the radiation detector. Collimated beams are generated by a synchrotron. A synchrotron produces monochromatic collimated coherent radiation with a high pho-ton flux, which avoids hardening artifacts and provides a high signal to noise ratio im-ages. The synchrotron radiation enables the selection of different wave lengths and the use of phase contrast images. The increasing brilliance of X-ray sources available nowa-days gives via the tomographic approach access to the fourth dimension: the time [Rack et al., 2010, Mokso et al., 2013, Maire and Withers, 2014, Di Michiel et al., 2005, Zabler et al., 2012]. This enables the application of time-resolved ➭CT for scanning processes. Synchrotron radiation is exclusively used in this work.

Synchrotron Light Source

The 3D images used in this thesis are produced at the European Synchrotron Radiation Facility (ESRF) on beam line ID19. A synchrotron consists of three parts shown in Fig. 1.3. At first, the electrons will be emitted and accelerated to near-light speed in the linear accelerator. Afterwards, the electron energy will be further increased to 6 GeV in the booster synchrotron. In the third step, the electrons are passed to the storage ring [Facility (ESRF), 2009]. Due to the high speed of electrons in the storage ring, relativistic effects take place and the low speed dipole radiation of electrons becomes to a narrow cone radiation pointing to the traveling direction [Schleede, 2013] and causing a high collimation.

Phase Contrast and Phase Retrieval

The absorption contrast is typically used in X-ray tomography. In the case investigated in this thesis, the cell simulators and the porous media are too small to weaken the radiation significantly to get interpretable data. For this reason phase contrast imaging is used. The phase shift is caused by a higher phase velocity in a medium represented by a complex refractive index n = 1 − δ − iβ (1.1). The real part 1 − δ stands for the elastic Rayleigh scattering caused by a forced oscilla-tion of bounded electrons, which are reemitting the forcing wave [Pedrotti et al., 2005]. The inelastic Compton scattering accounts for the imaginary part −β. It is caused by electrons, which absorb a part of the primary radiation and scatter a lower powered radiation. The refractive index is wave length dependent, whereas the material has an absolute refractive index n > 1 for visible light, the factor decreases to values smaller one for high energetic radiation. This causes a defocusing of a beam passing a convex body. Phase contrast is only available for spatial coherent radiation. Such radiations are produced by synchrotrons. Let us consider a single ray parallel to the x3-axis and passing through the medium with the time and spatial depending energy E(x3, t). In such condition the wave equation E(x3, t) = Eˆei(ωt−nk0 x3 ) = Eˆei(ωt−k0 x3 +δk0 x3 +iβk0 x3 ) (1.2).

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Characteristics of Random Fiber Fleeces

As illustrated in Fig. 2.3, fiber systems are non-woven fabrics made from continuous fibers or of fibers with limited length.
Let Φ be a random system of rectifiable space curves as defined in [Mecke and Nagel, 1980, Nagel, 1983]. In this setting, a random fiber fleece is the parallel set Ξ = Φ ⊕ Br , where r > 0 is the fiber radius, and Φ forms the random system of fiber cores. In order to ensure non-overlapping and smoothness of the fibers, it is supposed that there is a ε > 0 such that the fiber system Ξ is morphological closed with respect to a ball Bε of radius ε, (Ξ•Bε)=Ξ, [Wirjadi et al., 2016]. Realizations of macroscopic homogeneous random fiber fleece Ξ can be seen as a geometric model of fiber fleeces with circular cross sections of the fibers and a constant fiber diameter 2r. The distribution of the fiber fleece is invariant with respect to rotations around the z-axis. Besides porosity 1 − VV and surface density SV , the specific fiber length and the direction distribution are important characteristics. The specific fiber length can be estimated with the integral of mean curvature formulated in Eq. (2.5) divided by π. Interestingly, this also holds for curved fibers and for fibers with an arbitrary cross section.
There are numerous approaches to estimate the direction distribution of random fiber fleeces. One of the first estimation approaches deals with the cross section of the fibers produced by planar sections through the material [Zhu et al., 1997]. Obviously, the in-tersection of a plane with a cylinder produces an ellipse apart from boundary effects. The aspect ratio of the ellipse depends on the angle between the cylinder and the in-tersection plane. However, this aspect ratio does not uniquely identify one angle. In the paper [Wirjadi et al., 2009] a solution based on anisotropic and prolate Gaussian filtering is presented. The main idea is that the filter responds to a convolution with anisotropic Gaussian filters gets maximal if the direction of the longest principal axis of the filter is aligned to the fiber direction. The drawback of this approach is the dependency be-tween accuracy and number of considered directions, which is limited by the runtime. A very elegant and faster way to estimate the direction distribution is based on the directed distance transform. The directed distance transform calculates the distance of each foreground pixel to the background regarding all spatial directions given by the maximal pixel adjacency. This distance information is used to get the chord length for each direction in the fiber, which is assigned to each pixel of the chord [Altendorf and Jeulin, 2009]. Now, the inertia moments and axes of the chords can be determined and the eigenvalues of the inertia matrices can be calculated. The eigenvector having the lowest eigenvalue is indicating the fiber direction [Altendorf and Jeulin, 2009]. Neverthe-less, the computation effort for 3D images remains high. Shorter computation time can be reached when estimating the local fiber direction as the eigenvector belonging to the longest eigenvalue of the local Hessian matrix of the smoothed image f ∗ gσ [Tankyevych et al., 2009, Wirjadi et al., 2016], where gsσ is a Gaussian kernel of width σ.

Table of contents :

Chapter 1 Chromatographic Cell Separation 
1.1 Medical Context of Chromatography
1.2 Chromatographic Filters
1.3 Chemical Activation of the Inner Surface
1.4 Hydrodynamic Flow through Pore Space
1.5 Investigation of Chromatographic Filters by Tomography
1.5.1 Synchrotron Light Source
1.5.2 Phase Contrast and Phase Retrieval
1.5.3 The Inverse Radon Transform
1.6 General Aims of the Thesis
Chapter 2 Characteristics of Porous Media 
2.1 Foundations of Integral and Stochastic Geometry
2.1.1 Characteristics of Objects
2.1.2 Basic Characteristics of Random Structures
2.2 Characteristics of Porous Media
2.2.1 General Characteristics
2.2.2 Characteristics of Open Foams
2.2.3 Characteristics of Random Fiber Fleeces
2.3 Particle Paths
2.3.1 Characteristics of Continuous Space Curves
2.3.2 Discretization
2.3.3 Spline Approximation
2.3.4 Previous Work for Torsion Estimation
2.4 Scientific Challenge
Chapter 3 Curvature and Torsion of Particle Paths 
3.1 Discretization Schemes
3.1.1 Outer Jordan Discretization
3.1.2 Sequence of Particle Positions
3.2 Simulation of Particle Paths
3.2.1 Path of Fast Particles
3.2.2 Local Pixel Configurations
3.3 Fourier Approximation
3.4 Discretization of Differential Geometric Formulas
3.5 Estimation of the Third Derivatives of Real Functions – a Case Study
3.6 Conclusion
Chapter 4 Evaluation of the Estimation Methods 
4.1 Space Curves Examples
4.2 Evaluation of the Fourier Approximation
4.2.1 Boundary Effects
4.2.2 The Influence of Changes of Local Curvature and Torsion along the Curve
4.2.3 Fourier Analysis of the Discretization Noise
4.3 Evaluation of the Discretization of the Differential-Geometric Formulas
4.4 A Comparison Study
4.4.1 On the Choice of Smoothing Parameters
4.4.2 The Influence of Sampling
4.5 Conclusion
Chapter 5 Experimental Results 
5.1 Experimental Setup
5.1.1 Setup of Static Experiments
5.1.2 Setup of Dynamic Experiments
5.1.3 Time-resolved Microcomputed Tomography
5.2 Characterization of Partially Open Foams
5.2.1 Paths of Fast Particles in 3D Images
5.2.2 Path of Slow Particles
5.3 Characterization of Fiber Fleeces
5.4 A Random System of Overlapping Fibers
5.5 Conclusion
Conclusion and Perspectives 
Appendix A 3D Data Summary 
A.1 Dry Fleeces with Cells
A.2 Dry Partially Open Foams in Air
A.3 Partially Open Foams with Si Particle Suspension
A.4 Laminography Data
A.5 Fibers of Filter Cake


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