Fractal descriptors for quantifying brain complexity in MRI

Magnetic Resonance Imaging (MRI) is the in vivo technique most commonly employed to characterize changes in brain structures. The conventional MRI-derived morphological indices are able to capture only partial aspects of brain structural complexity. Fractal geometry and its most popular index, the fractal dimension (FD), can characterize self-similar structures including grey matter (GM) and white matter (WM). Previous literature shows the need for a definition of the so-called fractal scaling window, within which each structure manifests self-similarity. This justifies the existence of fractal properties and confirms Mandelbrot’s assertion that "fractals are not a panacea; they are not everywhere". In this work, we propose a new approach to automatically determine the fractal scaling window, computing two new fractal descriptors, i.e., the minimal and maximal fractal scales (mfs and Mfs). Our method was implemented in a software package, validated on phantoms and applied on large datasets of structural MR images. We demonstrated that the FD is a useful marker of morphological complexity changes that occurred during brain development and aging and, using ultra-high magnetic field (7T) examinations, we showed that the cerebral GM has fractal properties also below the spatial scale of 1 mm. We applied our methodology in two neurological diseases. We observed the reduction of the brain structural complexity in SCA2 patients and, using a machine learning approach, proved that the cerebral WM FD is a consistent feature in predicting cognitive decline in patients with small vessel disease and mild cognitive impairment. Finally, we showed that the FD of the WM skeletons derived from diffusion MRI provides complementary information to those obtained from the FD of the WM general structure in T1-weighted images. In conclusion, the fractal descriptors of structural brain complexity are candidate biomarkers to detect subtle morphological changes during development, aging and in neurological diseases.

Aspetti innovativi per la valutazione e la misura della porosità nei materiali dell'edilizia antica e moderna. La geometria frattale della porosità

New concepts on porosity appraisal in ancient and modern construction materials. The role of Fractal Geometry on porosity characterization and transport phenomena. This work studied the potential of Fractal Geometry to the characterization of porous materials. Besides the descriptive aspects of the pore size distribution, the fractal dimensions have led to the development of rational relations for the prediction of permeability coefficients to fluid and heat transfer. The research considered natural materials used in historical buildings (rock and earth) as well as currently employed materials as hydraulic cement and technologically advanced materials such as silicon carbide or YSZ ceramics. The experimental results of porosity derived from the techniques of mercury intrusion and from the image analysis. Data elaboration was carried out according to established procedures of Fractal Geometry. It was found that certain classes of materials are clearly fractal and respond to simple patterns such as Sierpinski and Menger models. In several cases, however, the fractal character is not recognised because the microstructure of the material is based on different phases at different dimensional scales, and in consequence the “fractal dimensions” calculated from porosimetric data do not come within the standard range (less than 3). Using different type and numbers of fractal units is possible, however, to obtain “virtual” microstructures that have the fraction of voids and pore size distribution equivalent with the experimental ones for almost any material. Thus it was possible to take the expressions for the permeability and the thermal conduction which does not require empirical “constants”, these expressions have also provided values that are generally in agreement with the experimental available data. More problematic has been the fractal discussion of the geometry of the rupture of the material subjected to mechanical stress both external and internal applied. The results achieved on these issues are qualitative and prone to future studies. Keywords: Materials, Microstructure, Porosity, Fractal Geometry, Permeability, Thermal conduction, Mechanical strength.

Theoretical studies of microfluidics: "walking" drops and "melting" snowflakes.

The present PhD thesis summarizes two examples of research in microfluidics. Both times water was the subject of interest, once in the liquid state (droplets adsorbed on chemically functionalized surfaces), the other time in the solid state (ice snowflakes and their fractal behaviour). The first problem deals with a slipping nano-droplet of water adsorbed on a surface with photo-switchable wettability characteristics. Main focus was on identifying the underlying driving forces and mechanical principles at the molecular level of detail. Molecular Dynamics simulation was employed as investigative tool owing to its record of successfully describing the microscopic behaviour of liquids at interfaces. To reproduce the specialized surface on which a water droplet can effectively “walk”, a new implicit surface potential was developed. Applying this new method the experimentally observed droplet slippage could be reproduced successfully. Next the movement of the droplet was analyzed at various conditions emphasizing on the behaviour of the water molecules in contact with the surface. The main objective was to identify driving forces and molecular mechanisms underlying the slippage process. The second part of this thesis is concerned with theoretical studies of snowflake melting. In the present work snowflakes are represented by filled von Koch-like fractals of mesoscopic beads. A new algorithm has been developed from scratch to simulate the thermal collapse of fractal structures based on Monte Carlo and Random Walk Simulations (MCRWS). The developed method was applied and compared to Molecular Dynamics simulations regarding the melting of ice snowflake crystals and new parameters were derived from this comparison. Bigger snow-fractals were then studied looking at the time evolution at different temperatures again making use of the developed MCRWS method. This was accompanied by an in-depth analysis of fractal properties (border length and gyration radius) in order to shed light on the dynamics of the melting process.

Analisi d'immagine, patologia quantitativa e dimensione frattale in patologia veterinaria: Densita' e forme

Fino dagli albori della metodica scientifica, l’osservazione e la vista hanno giocato un ruolo fondamentale. La patologia è una scienza visiva, dove le forme, i colori, le interfacce e le architetture di organi, tessuti, cellule e componenti cellulari guidano l’occhio del patologo e ne indirizzano la scelta diagnostico-classificativa. L’osservazione del preparato istologico in microscopia ottica si attua mediante l’esame e la caratterizzazione di anomalie ad ingrandimenti progressivamente crescenti, a diverse scale spaziali, che partono dalla valutazione dell’assetto architettonico sovracellulare, per poi spostarsi ad investigare e descrivere le cellule e le peculiarità citomorfologiche delle stesse. A differenza di altri esami di laboratorio che sono pienamente quantificabili, l’analisi istologica è intrinsecamente soggettiva, e quindi incline ad un alto grado di variabilità nei risultati prodotti da differenti patologi. L’analisi d’immagine, l’estrazione da un’immagine digitale di contenuti utili, rappresenta una metodica oggettiva, valida e robusta ormai largamente impiegata a completamento del lavoro del patologo. Si sottolinea come l’analisi d’immagine possa essere vista come fase descrittiva quantitativa di preparati macroscopici e microscopici che poi viene seguita da una interpretazione. Nuovamente si sottolinea come questi descrittori siano oggettivi, ripetibili e riproducibili, e non soggetti a bassa concordanza inter operatore. La presente tesi si snoda attraverso un percorso concettuale orientato ad applicazioni di analisi d’immagine e patologia quantitativa che parte dalle applicazioni più elementari (densità, misure lineari), per arrivare a nozioni più avanzate, quali lo studio di complessità delle forme mediante l’analisi frattale e la quantificazione del pattern spaziale di strutture sovracellulari.