Obtaining microstructural information non-invasively on brain tissue remains a challenge. Diffusion magnetic resonance imaging (dMRI) is an imaging method that can provide such information. That includes geometrical considerations of nerve cells projections, axons, that are present in the white matter of the human brain. Axons carry information encoded into electrical impulses to other cells. The thesis deals with estimating parameters of the axonal trajectories, modeled as one-dimensional pathways, from the dMRI signal. That is achieved in two steps: constructing a forward model to predict the dMRI signal and, vice versa, estimating the tissue parameters from dMRI signal by solving the so-called inverse problem. The proposed forward model employs a spectral analysis of dMRI signal. This formulation enables signal prediction for any gradient waveform and helps to identify the physical characteristics of the underlying system that are preserved in the dMRI signal. The physical properties are represented in so-called diffusion spectra whereas gradient waveforms, that sensitizes the signal, are in the encoding spectra. To mimic biologically plausible axonal trajectories, axonal trajectories were modeled by a 1D-toy model that incorporates harmonic waves with variable degree of randomness. Different numerical methods for computation of diffusion spectra were compared, and the resulting spectra were characterized by a phenomenological model incorporating three parameters. It was not possible to estimate the exact parameters of the 1D-toy model from diffusion spectra. Nonetheless, it was possible to estimate their statistical descriptors, namely microscopic orientation dispersion and dispersion-weighted wavelength. Solving the inverse problem posed a major challenge. The phenomenological model of the diffusion spectra was incorporated in a forward model of the diffusion-weighted signal perpendicular to the trajectory and applied to a state-of-the-art data acquired in human brain white matter of a healthy volunteer. It was not possible to estimate all the parameters of the phenomenological model but by constraining the parameters to plausible values we could estimate the last that was within the range predicted by histology. Incorporating trajectory-parameters in the model of white matter diffusion yielded fit residuals as small as those obtained with current state-of-the-art models assuming parallel, straight, and cylindrical cylinders. However, the cylinder model predicted axon diameters far outside the range expected from histology. We conclude that neglecting the axonal trajectories leads to biased models of axons in brain white matter.MRI can serve as an example of successfully applied fundamental research from physics to biological sciences, humanities, chemistry or medicine. Biomolecules in biochemistry can be probed with atomic resolution. Nanomaterials in material sciences, porous rocks in geology, cell structures or tissues in biology and medicine can be examined. Statistical analysis of MRI signal can reveal functional state of the brain and is relevant in e.g. psychology. This thesis deals mainly with applications within medical sciences. Diffusion magnetic resonance imaging (dMRI) unravels the tissue microstructure, i.e. the structure of tissue on the micrometer length scale. At this scale, the arrangement of cells and other biologically relevant structures emerges as a new property from a deeper, biochemical, scale. Microstructural appearance is often defining feature of biological tissues and is intertwined with their biological behavior, which is a highly interesting information from a medical point of view. In this project, we study in a systematic way, often neglected, geometrical aspects of axons called axonal trajectories. Axons are the wiring of the brain. Based on microscopical images we proposed their representation, inspected their properties and forecasted the outcome of a diffusion measurement. The inverse question, whether the information on the axonal trajectories can be inferred from the outcome of measurement, and whether they could be neglected, was answered as well. The results suggest that non-straight axonal trajectories need to be considered in the of representations of axons, although to estimate them reliably the practical diffusion measurements need to be improved. The estimated properties of axonal trajectories were congruent to the gold-standard method, microscopy. Same methodology applied to the investigation of axonal trajectories can be employed in other problems in the dMRI field and may also lead to better understanding of the nature of the results of the diffusion measurements in the human tissue. Potentially, novel biomarkers that could help to diagnose diseases could be discovered. Generally, dMRI is an interesting research field where potential breakthrough could be made. It probes the microstructural region that is highly important from the biological point of view, has a solid foundation in physical theory, allows for large variety of possible arrangements of the dMRI experiments and is not as widespread as other imaging modalities
