1,721,031 research outputs found
Inference for the mean of large p small n data: a high dimensional generalization of Hotelling’s theorem
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Hotelling’s T2 in separable Hilbert spaces
No full textWe address the problem of finite-sample null hypothesis significance testing on the mean element of a random variable that takes value in a generic separable Hilbert space. For this purpose, we propose a (re)definition of Hotelling’s T2
that naturally expands to any separable Hilbert space that we further embed within a permutation inferential approach. In detail, we present a unified framework for making inference on the mean element of Hilbert populations based on Hotelling’s T2
statistic, using a permutation-based testing procedure of which we prove finite-sample exactness and consistency; we showcase the explicit form of Hotelling’s T2
statistic in the case of some famous spaces used in functional data analysis (i.e., Sobolev and Bayes spaces); we demonstrate, by means of simulations, that Hotelling’s T2
exhibits the best performances in terms of statistical power for detecting mean differences between Gaussian populations, compared to other state-of-the-art statistics, in most simulated scenarios; we propose a case study that demonstrate the importance of the space into which one decides to embed the data; we provide an implementation of the proposed tools in the R package fdahotelling available at https://github.com/astamm/fdahotelling
A personalized mathematical tool for neuro-oncology: A clinical case study
No full textThis work evaluates the predictive ability of a novel personalized computational tool for simulating the growth
of brain tumours using the neuroimaging data collected during one clinical case study. The mathematical model
consists of an evolutionary fourth-order partial differential equation with degenerate motility, in which the
spreading dynamics of the multiphase tumour is coupled with a parabolic equation determining the diffusing
oxygen within the brain. The model also includes a reaction term describing the effects of radiotherapy, that
is simulated in accordance with the clinical schedule. We collect Magnetic Resonance (MRI) and Diffusion
Tensor (DTI) imaging data for one patient at given times of key clinical interest, from the first diagnosis of
a giant glioblastoma to its surgical removal and the subsequent radiation therapies. These neuroimaging data
allow reconstructing the patient-specific brain geometry in a finite element virtual environment, that is used for
simulating the tumour recurrence pattern after the surgical resection. In particular, we characterize the different
brain tissues and the tumour location from MRI data, whilst we extrapolate the heterogeneous nutrient diffusion
parameters and cellular motility from DTI data.
The numerical results of the simulated tumour are found in good qualitative and quantitative agreement with
the volume and the boundaries observed in MRI. Moreover, the simulations point out a consistent regression of
the tumour mass in correspondence to the application of radiotherapy, with an average growth rate which is of
the same order as the one calculated from the neuroimaging data. Remarkably, our results display the highest
Jaccard index of the tumour region reported in the biomathematical literature. In conclusion, this work represents
an important proof-of-concept of the ability of this mathematical framework to predict the tumour recurrence
and its response to therapies in a patient-specific manne
CEREBRO : CrEation of REalistic Brains for dmRi inspectiOn
No full textLAUREA MAGISTRALETra le varie tecniche esistenti di ispezione del corpo umano, la Risonanza Magnatica (MR) è l'unica ad essere in-vivo e non invasiva. La tecnica di imaging più utilizzata per generare contrasto nelle immagini MR è la Risonanza Magnetica di diffusione (dMRI) che sfrutta la diffusione dell'acqua naturalmente abbondante nei tessuti per ricostruirne la microstruttura. La dMRI trova la sua principale applicazione al cervello umano ed in particolare alla Materia Bianca che è composta da densi pacchetti di assoni, chiamati fascicles, che non sono visibili ad occhio nudo.
La dMRI ha due grosse componenti, caratterizzate dai modelli di rumore e diffusione, che vengono spesso considerate in maniera autonoma. In questo lavoro, per la prima volta, cercheremo di considerarle entrambe con la stessa importanza.
Inoltre, data la grande importanza di ricostruire correttamente la microstruttura della Materia Bianca, abbiamo sviluppato cerebro, grazie al software statistico R, che è in grado di generare datasets realistici del segnale dMRI applicato ad una porzione 2D del cervello umano.
La maggior parte degli algoritmi esistenti per fare simulazioni in questo ambito ha il grande svantaggio di non considerare la mappa di sensitività delle bobine che generano il segnale. Proponiamo quindi un simulatore alternativo che ne tenga conto e che fornisca anche degli esempi realistici come dati del pacchetto. E' stata messa molta enfasi nei calcoli e nel fissare i parametri in modo tale da farli risultare più realistici possibile.
In aggiunta, i datasets generati possono essere usati per ulteriori analisi dato che cerebro è stato pensato con l'intenzione di supportare altri strumenti operanti nel settore dMRI. In particolare, dato che questo campo abbonda di assunzioni e modelli, si potrebbe veri care se il Modello Gaussiano funziona bene o se bisognerebbe assumere un modello nc-χ in qualche scenario.Among the existing techniques of inspection of the human body, the Magnetic Resonance (MR) is the only one that is in-vivo and non invasive. The most used imaging technique to generate contrast in MR images is the Diffusion Magnetic Resonance Imaging (dMRI). It exploits the diffusion of the water naturally present in the tissues in order to reconstruct the underlying microstructure. One of its main application is the human brain and in particular the White Matter (WM) that is mainly composed by dense bundles of axons, called fascicles that are not visible with naked eye.
The dMRI eld has two main components characterized by the diffusion and noise models that are always treated separately. In this work, for the first time, we try to consider them both with the same importance.
Moreover, given the great importance of correctly reconstructing the WM microstructure, we propose cerebro, developed thanks to the open-source statistical software R, that is able to generate realistic datasets coming from the dRMI signal of a 2D slice of human brain.
The majority of existing algorithms and tools for doing simulations in this field have the big disadvantage of not considering the sensitivity map of the coils generating the signal. Thus we propose an alternative simulator that takes it into account and we provide also realistic examples as data of the package. We put a lot of effort in calculating them and tuning the different parameters in order to make them as much realistic as possible.
In addition, the generated datasets can be used for further analysis since we developed cerebro with the main intention of supporting other tools working in the dMRI field. In particular, since the dMRI abounds with assumptions and models, it can be used for testing if the widely used Gaussian model works well or if in some framework it would be better to assume a nc-χ model
MASTER : a multivariate axonal simulator for tractography e valuation in R towards reliable reconstruction of brain white matter fascicles
No full textLAUREA MAGISTRALEOur brain receives and processes information via neurons, which are cells connected together by axons, grouped into dense bundles called fascicles. When these latter are damaged, parts of the body functions might be defective, which might lead to fatal outcomes for patients. However, the axonal cytoarchitecture of the brain is not visible to the naked eye. It is thus of critical importance for the diagnosis and treatment of brain disorders to depict as accurately as possible and in the most reproducible fashion this cytoarchitecture. Using mathematical models, for instance, brain tumor removal surgery would be greatly improved if such reconstructions could be reliably provided. Diffusion MRI enables in-vivo and non-invasive reconstruction of such fascicles. However this reconstruction is prone to external influences, such as imaging artifacts and noise; moreover it involves a number of non-trivial modeling steps which have not been successfully validated yet. This motivated the realization of both hardware and software frameworks, on which to test reconstruction algorithms, which unfortunately are not able to correctly deal with complex configurations of fascicles or, alternatively, lack ground truth. There is thus an urgent unmet need for a validation tool able to correctly reproduce the white matter cytoarchitecture. To overcome this problem we propose master, a Multivariate Axonal Simulator for Tractography Evaluation in R, which enables to simulate the white matter fascicles configurations. master has been developed with the open-source statistical software R. It is the result of a deep study of the white matter cytoarchitecture, which brought us to innovatively reproduce each fascicle configuration from a single simple mathematical model with few, and very intuitive, input parameters that are relevant to a reliable cytoarchitecture reconstruction. master is available both as R-package and as web application; moreover its portability is enhanced thanks to the possibility to export simulated data as .csv files
Non-parametric classification and regression techniques for the characterisation of the disease subtypes and the assessment of the temporal evolution of image-based biomarkers
Full text linkIn questa tesi, si presentano alcuni modelli e metodi statistici non parametrici, sviluppati e adattati per gestire diversi tipi di biomarcatori. In particolare, si descrive la stima dell'evoluzione della funzione respiratoria dalla fanciullezza all'età adulta di pazienti affetti dalla Distrofia Muscolare di Duchenne, dove le misurazioni sono state acquisite longitudinalmente a tempi irregolari e specifici per ogni soggetto. In questo caso, si adotta un modello di regressione a effetti misti basato su spline cubiche, che permette di identificare specifici istanti temporali di peggioramento della funzione respiratoria durante la progressione della malattia, e di investigare possibili effetti della scoliosi, della ventilazione meccanica notturna non invasiva e della terapia steroidea. Nel seguito della tesi, si caratterizzano i sottotipi della malattia di Creutzfeldt-Jakob sporadica con biomarcatori da imaging medico, acquisiti con campionamento cross-sectional. In questo caso, i biomarcatori considerati sono le iper-intensità del segnale di diffusione tramite imaging a risonanza magnetica, le quali sono misurate con un sistema semi-qualitativo in alcune regioni cerebrali. Quindi, si classificano i pazienti nel sottotipo più compatibile della malattia di Creutzfeldt-Jakob sporadica, secondo le loro misurazioni dei biomarcatori, attraverso un metodo basato sugli alberi di classificazione. Inoltre, si descrive la progressione della malattia in ognuno di tali sottotipi, identificando la sequenza delle regioni cerebrali che diventano distinguibilmente iperintense nelle immagini di risonanza magnetica pesate in diffusione. Per raggiungere tale obiettivo, si adatta il cosiddetto "event-based model" recentemente introdotto in letteratura, che consiste in un modello statistico data-driven che stima l'evoluzione della malattia in termini dei suoi biomarcatori caratterizzanti, senza basarsi su un dataset longitudinale. Nella parte finale della tesi, si delinea un lavoro che vuole sviluppare un modello di regressione "function-on-function", che possa gestire biomarcatori con dipendenza temporale (ad esempio, immagini ottenute tramite risonanza magnetica funzionale). A tale scopo, si modellizza la risposta funzionale in termini di diverse covariate funzionali e si propone un test basato su permutazioni per identificare sotto-regioni che esibiscono differenze statistiche simili. Inoltre, nel caso di test multipli effettuati in diversi punti dello stesso dominio (ad esempio, i voxel dell'immagine cerebrale ottenuta mediante risonanza magnetica), si estende a un contesto tridimensionale il "closure multiplicity adjustment method" per controllare il family-wise error rate della procedura proposta.In this thesis, we present some non-parametric statistical models and methods that have been developed and adapted to deal with different types of biomarker. In particular, we describe the assessment of the respiratory function evolution of Duchenne Muscular Dystrophy (DMD) patients from childhood to adulthood, where measurements are collected longitudinally at irregular and subject-specific times. We adopt here a regression model based on natural cubic splines with mixed effects, that allows to identify specific time points of respiratory impairment during disease progression, and to investigate possible effects of scoliosis, nocturnal non-invasive mechanical ventilation and steroid therapy. Then, we characterise the sybtypes of the sporadic Creutzfeldt-Jakob disease (sCJD) with imaging biomarkers collected in a cross-sectional design. In this case, the considered biomarkers are the signal hyperintensities of diffusion magnetic resonance imaging (dMRI), that are measured with a semi-quantitative scoring system devised to visually assess the images in different brain regions. We classify the sCJD patients into their most compatible subtype according to their biomarker measurements, with a classification tree-based method. Moreover, we describe the disease progression in each sCJD subtype by finding the sequence of brain regions that become detectably hyperintense on the diffusion images. We adapt the recently introduced event-based model, a data-driven statistical model that assess the disease evolution in terms of its characterising biomarkers, without relying on a longitudinal dataset. Finally, we outline a work aimed at developing a function-on-function regression model that can deal with temporal dependent biomarkers (e.g., from functional magnetic resonance imaging data). We model a functional response in terms of several functional covariates, and we propose a permutation test to identify sub-regions that exhibit similar statistical differences. Moreover, in case of multiple tests performed at different locations in the same domain (e.g., the voxels of the brain MR image), we extend to a three-dimensional setting the closure multiplicity adjustment method to control the family-wise error rate of the proposed procedure.DIPARTIMENTO DI MATEMATICA30LUCCHETTI, ROBERTOSABADINI, IRENE MARI
Optimized K-mean alignment algorithm for clustering functional data : application to brain tractography
No full textLAUREA MAGISTRALEQuesto lavoro è stato sviluppato nel ambito dell’analisi di dati funzionali. In
particolare viene preso in considerazione l’algoritmo K-mean Alignment che
contemporaneamente registra e classifica dati funzionali. Dopo l’introduzione
di un rigoroso framework teorico, l’algoritmo è descritto ed efficientemente
implementato per poter essere applicato a datasets di grandi dimensioni.
Questa implementazione è disponibile su GitHub sotto forma di un pacchetto R chiamato fdakmapp. Inoltre, un’applicazione web è disponibile per
esplorare le funzionalità del pacchetto con diversi datasets. Nella seconda parte della tesi l’algoritmo è stato applicato a dei dati che descrivono
il tratto corticospinale (CST) nel cervello di 20 pazienti sani. Il CST è un
tratto critico della materia bianca perché collega la corteccia motoria primaria con il midollo spinale essendo responsabile del movimento volontario.
In letteratura esistono atlanti rappresentanti le connessioni celebrali più importanti ma sorprendemente mancano atlanti che descrivono la localizzazione
dei specifici tratti. In questo lavoro proponiamo una metodologia statistica
completa per costruire un atlante dei CST sani. Questo sarà utile nell’ ideare
tests statistici per rilevare tessuti danneggiati lungo il CST e di conseguenza
migliorare l’esito dei trattamenti per pazienti con malattie neurologiche (tumori, ictus, Parkinson e disturbi correlati, etc.).The present work is developed in the field of functional data analysis. In particular, it focuses on the K-mean Alignment algorithm that jointly performs continuous registration and clustering. After an introduction of a proper theoretical framework, the algorithm is described and efficiently implemented in
order to be used with large datasets. The efficient implementation is publicly
available on GitHub in the form of an R package called fdakmapp. Moreover,
a web application is provided to test the package functionalities also with
custom datasets. In the second part of the work, the algorithm is applied to
a set of data describing the Corticospinal Tract (CST) in the brains of 20
subjects. The CST is a critical white matter tract as it connects the primary
motor cortex to the spinal cord and handles voluntary motion. Atlases of
major brain connections do exist, but, surprisingly, atlases that depict the
actual localization of a specific pathway are missing. In this work, we propose a comprehensive statistical methodology for generating an atlas of the
healthy CST. This should help to design efficient statistical tests for detecting
damaged tissue along the CST and consequently improving patient outcome
in some brain pathologies (tumors, strokes, Parkinson and related disorders,
etc.)
Hotelling in wonderland
No full textWhile Hotelling’s T2 statistic is traditionally defined as the Mahalanobis
distance between the sample mean and the true mean induced by the inverse of the
sample covariance matrix, we hereby propose an alternative definition which allows
a unifying and coherent definition of Hotelling’s T2 statistic in any Hilbert space
independently from its dimensionality and sample size. In details, we introduce the
definition of random variables in Hilbert spaces, the concept of mean and covariance
in such spaces and the relevant operators for formulating a proper definition of
Hotelling’s T2 statistic relying on the concept of Bochner integral
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