A nonconservative kinetic model under the action of an external force field for modeling the medical treatment of autoimmune response

In this paper, we develop a new nonconservative kinetic framework to be applied for the study of immune system dysregulation. From the modeling viewpoint, the new kinetic model regards a system, composed of stochastically interacting agents, under the action of an eternal force field. According to the application perspectives of this paper, the external force field has a specific analytical shape. In this case, some analytical results are proved, i.e. existence, uniqueness, positivity and boundedness of solution of the related Cauchy problem, under suitable assumptions, at least locally in time. Then, the models is refined to be implemented for the study of treatment strategies in case of autoimmune response. Specifically, we distinguish the autonomous case from the nonautonomous one, representing absence or delivery of drugs, respectively. The former allows to gain some stability results, still in terms of Hopf bifurcations. Whereas, the latter is qualitatively studied. Numerical simulations are provided for both schemes. They show the adherence of this new framework to the realistic scenarios, confirming its impact and potentiality

Reaction-diffusion systems derived from kinetic models for Multiple Sclerosis

We present a mathematical study for the development of Multiple Sclerosis in which a spatio-temporal kinetic model describes, at mesoscopic level, the dynamics of a high number of interacting agents. We consider both interactions among different populations of human cells and motion of immune cells, stimulated by cytokines. Moreover, we reproduce the consumption of myelin sheath due to anomalously activated lymphocytes and its restoration by oligodendrocytes. Successively, we fix a small time parameter and assume that the considered processes occur at different scales. This allows to perform a formal limit, obtaining macroscopic reaction-diffusion equations for the number densities with a chemotaxis term. A natural step is then to study the system, inquiring about the formation of spatial patterns through a Turing instability analysis of the problem and basing the discussion on microscopic parameters of the model. In particular, we get spatial patterns oscillating in time that may reproduce brain lesions characteristic of different phases of the pathology

Optimal control of leachate recirculation for anaerobic processes in landfills

A mathematical model for the degradation of the organic fraction of solid waste in landfills, by means of an anaerobic bacterial population, is proposed. Additional phenomena, like hydrolysis of insoluble substrate and biomass decay, are taken into account. The evolution of the system is monitored by controlling the effects of leachate recirculation on the hydrolytic process. We investigate the optimal strategies to minimize substrate concentration and recirculation operation costs. Analytical and numerical results are presented and discussed for linear and quadratic cost functionals

Modelli BGK ed equazioni di reazione-diffusione per miscele reagenti di gas monoatomici e poliatomici

Lo studio delle miscele di gas è un tema che oggigiorno risponde alle necessità di vari campi di ricerca, come l'ingegneria aerospaziale, gli studi climatici, le industrie energetiche, ecc. Per questo motivo la costruzione di modelli matematici che simulino il comportamento di gas reali si rivela estremamente utile. Tra tutti gli approcci possibili, quello cinetico, che si basa sulle equazioni di Boltzmann per le funzioni di distribuzione dei gas, rappresenta uno degli strumenti più validi. Esso permette, infatti, di descrivere miscele a partire dall'interazione tra particelle, per poi derivare modelli per le quantità osservabili. Il lavoro contenuto in questa tesi è volto a riprendere i risultati presenti in letteratura per miscele di gas e a estenderli considerando casi più realistici, come miscele di specie monoatomiche e poliatomiche, che interagiscono in modo inelastico o chimico. Per prima cosa, nell'introduzione vengono presentati i concetti basilari e i risultati più rilevanti per lo studio cinetico dei gas, insieme a una sintesi più dettagliata dei contenuti della tesi. Nel Capitolo 1 proponiamo lo studio di una miscela reattiva costituita da quattro gas utilizzando la classica teoria cinetica di Boltzmann. Questo problema è già stato analizzato nell’ipotesi in cui i gas hanno lo stesso numero di livelli di energia interna, lo affrontiamo nel caso più generale supponendo che ciascuna delle specie coinvolte abbia un diverso numero di livelli energetici. Nei due capitoli successivi vengono studiate miscele di gas utilizzando un approccio cinetico di tipo BGK. In particolare, nel Capitolo 2 forniamo un modello BGK per una miscela inerte di gas monoatomici e poliatomici. Dimostriamo la consistenza del modello e analizziamo la stabilità degli equilibri; deriviamo poi opportune equazioni macroscopiche ed eseguiamo alcune simulazioni numeriche ispirandoci ai gas reali. Nel Capitolo 3, invece, proponiamo due modelli BGK per miscele di gas reagenti. Nel primo consideriamo quattro specie di gas coinvolte in una reazione chimica reversibile, nel secondo otto gas che partecipano a due reazioni disgiunte. La procedura precedente viene applicata in entrambi i casi, la principale differenza risiede nel dimostrare la consistenza del modello, poiché si ottengono equazioni trascendenti più complicate per la determinazione di tutti i parametri. Anche in questo contesto vengono eseguite simulazioni numeriche che modellino il comportamento di miscele reattive reali. Nella parte restante della tesi, studiamo miscele di gas mediante tecniche ulteriori. Nel Capitolo 4 consideriamo una miscela di cinque specie di gas, di cui tre costituiscono il mezzo ospite in cui interagiscono le altre due. Gli urti tra le particelle possono essere di tipo elastico, inelastico o chimico e ipotizziamo che questi avvengano su scale temporali diverse. Successivamente, scriviamo le equazioni di Boltzmann classiche per le funzioni di distribuzione delle varie componenti. Dopo opportune integrazioni delle equazioni e tramite un passaggio al limite otteniamo equazioni di reazione-diffusione per le densità di specie. Nello specifico, applichiamo questo procedimento in tre diversi regimi idrodinamici, ottenendo per ciascuno di essi un diverso sistema di reazione-diffusione. Le proprietà di stabilità di tali sistemi vengono discusse nel Capitolo 5. Ci concentriamo in particolare sul verificarsi del fenomeno dell’instabilità di Turing per scelte opportune dei valori energetici e delle frequenze di collisione. Attraverso simulazioni numeriche, verifichiamo poi la formazione di pattern nell’evoluzione delle densità, come previsto dall'analisi di Turing. Concludiamo con alcune ulteriori osservazioni e prospettive per futuri sviluppi del presente lavoro di ricerca.The study of gas flows is an issue that nowadays responds to the necessities of various fields of research, as aerospace engineering, climate studies, energy industries, etc. For this reason, the construction of mathematical models simulating the behavior of real gas mixtures is extremely useful. Among all possible approaches, the kinetic one, based on Boltzmann equations for species distribution functions, seems to be a very powerful tool. It allows, in fact, to describe mixtures starting from interaction among particles, with the possibility of deriving models for the behavior of the global system at observable level. The work of this thesis is devoted to considering results obtained so far for mixtures of gases and extending them considering more real-like cases, such as mixtures of monoatomic and polyatomic gas species, that may also interact inelastically or chemically. As first, we provide an introduction in which the basic concepts and the most relevant results for kinetic description of gases are presented, along with a more detailed summary of the work carried out in the thesis. In Chapter 1, we propose the study of a reacting mixture of four gases using the classical Boltzmann kinetic theory. This case was already analyzed when the four gases are considered to have the same number of internal energy levels. We generalize it allowing each of the gas species to have a different number of energetic levels. Chapter 2 and Chapter 3 are devoted to the study of gas mixtures using a kinetic approach of BGK type. In particular, in Chapter 2 we provide a BGK model for an inert mixture of monatomic and polyatomic gases. We prove the consistency of the model and analyze the stability of equilibria, then we derive macroscopic equations and perform some numerical simulations being inspired by real gases. In Chapter 3, instead, we propose two BGK models for mixtures of reacting gases. In the first one we have four gas species involved in a reversible chemical reaction, in the second case eight gases react through two disjoint reactions. The previous strategy is applied to both cases, the main differences are in proving the consistency of the model, since we face more complicate transcendental equations to determine all the parameters. Also in these cases, numerical simulations are performed to reproduce the behavior of real reacting mixtures. In the remaining part of the thesis, we study gas mixtures using different techniques. In Chapter 4 we consider a mixture of five gas species, three of them constituting a background medium in which the other two interact. Encounters among particles can be elastic, inelastic, or chemical and we suppose that they occur at different time scales. We write classical Boltzmann equations for the interacting components, we pass to the asymptotic diffusive limit and, by means of suitable integrations of the kinetic equations, we obtain reaction-diffusion equations for densities of the species. Specifically, we apply this procedure in three different hydrodynamic regimes, obtaining in each case a proper reaction-diffusion system. The stability properties of these systems are then studied in Chapter 5. We consider the possibility of having Turing instability for a suitable choice of internal energy amounts and of collision frequencies. Through numerical simulations, we verify the formation of spatial patterns in the evolution of the number densities of reactants, as predicted by Turing analysis. We conclude with some further observations and perspectives for a future development of the present research work

Superior Cerebellar Atrophy: An Imaging Clue to Diagnose <i>ITPR1</i>-Related Disorders

The inositol 1,4,5-triphosphate receptor type 1 (ITPR1) gene encodes an InsP3-gated calcium channel that modulates intracellular Ca2+ release and is particularly expressed in cerebellar Purkinje cells. Pathogenic variants in the ITPR1 gene are associated with different types of autosomal dominant spinocerebellar ataxia: SCA15 (adult onset), SCA29 (early-onset), and Gillespie syndrome. Cerebellar atrophy/hypoplasia is invariably detected, but a recognizable neuroradiological pattern has not been identified yet. With the aim of describing ITPR1-related neuroimaging findings, the brain MRI of 14 patients with ITPR1 variants (11 SCA29, 1 SCA15, and 2 Gillespie) were reviewed by expert neuroradiologists. To further evaluate the role of superior vermian and hemispheric cerebellar atrophy as a clue for the diagnosis of ITPR1-related conditions, the ITPR1 gene was sequenced in 5 patients with similar MRI pattern, detecting pathogenic variants in 4 of them. Considering the whole cohort, a distinctive neuroradiological pattern consisting in superior vermian and hemispheric cerebellar atrophy was identified in 83% patients with causative ITPR1 variants, suggesting this MRI finding could represent a hallmark for ITPR1-related disorders

Heterozygous KIF1A variants underlie a wide spectrum of neurodevelopmental and neurodegenerative disorders

Background: Dominant and recessive variants in the KIF1A gene on chromosome 2q37.3 are associated with several phenotypes, although only three syndromes are currently listed in the OMIM classification: hereditary sensory and autonomic neuropathy type 2 and spastic paraplegia type 30, both recessively inherited, and mental retardation type 9 with dominant inheritance. Methods: In this retrospective multicentre study, we describe the clinical, neuroradiological and genetic features of 19 Caucasian patients (aged 3-65 years) harbouring heterozygous KIF1A variants, and extensively review the available literature to improve current classification of KIF1A-related disorders. Results: Patients were divided into two groups. Group 1 comprised patients with a complex phenotype with prominent pyramidal signs, variably associated in all but one case with additional features (ie, epilepsy, ataxia, peripheral neuropathy, optic nerve atrophy); conversely, patients in group 2 presented an early onset or congenital ataxic phenotype. Fourteen different heterozygous missense variants were detected by next-generation sequencing screening, including three novel variants, most falling within the kinesin motor domain. Conclusion: The present study further enlarges the clinical and mutational spectrum of KIF1A-related disorders by describing a large series of patients with dominantly inherited KIF1A pathogenic variants ranging from pure to complex forms of hereditary spastic paraparesis/paraplegias (HSP) and ataxic phenotypes in a lower proportion of cases. A comprehensive review of the literature indicates that KIF1A screening should be implemented in HSP regardless of its mode of inheritance or presentations as well as in other complex neurodegenerative or neurodevelopmental disorders showing congenital or early onset ataxia

Refining the mutational spectrum and gene–phenotype correlates in pontocerebellar hypoplasia: results of a multicentric study

Background: Pontocerebellar hypoplasias (PCH) comprise a group of genetically heterogeneous disorders characterised by concurrent hypoplasia of the pons and the cerebellum and variable clinical and imaging features. The current classification includes 13 subtypes, with ~20 known causative genes. Attempts have been made to delineate the phenotypic spectrum associated to specific PCH genes, yet clinical and neuroradiological features are not consistent across studies, making it difficult to define gene-specific outcomes. Methods: We performed deep clinical and imaging phenotyping in 56 probands with a neuroradiological diagnosis of PCH, who underwent NGS-based panel sequencing of PCH genes and MLPA for CASK rearrangements. Next, we conducted a phenotype-based unsupervised hierarchical cluster analysis to investigate associations between genes and specific phenotypic clusters. Results: A genetic diagnosis was obtained in 43 probands (77%). The most common causative gene was CASK, which accounted for nearly half cases (45%) and was mutated in females and occasionally in males. The European founder mutation p.Ala307Ser in TSEN54 and pathogenic variants in EXOSC3 accounted for 18% and 9% of cases, respectively. VLDLR, TOE1 and RARS2 were mutated in single patients. We were able to confirm only few previously reported associations, including jitteriness and clonus with TSEN54 and lower motor neuron signs with EXOSC3. When considering multiple features simultaneously, a clear association with a phenotypic cluster only emerged for EXOSC3. Conclusion: CASK represents the major PCH causative gene in Italy. Phenotypic variability associated with the most common genetic causes of PCH is wider than previously thought, with marked overlap between CASK and TSEN54-associated disorders

Corpus callosum abnormalities: neuroimaging, cytogenetics and clinical characterization of a very large multicenter Italian series

Corpus callosum abnormalities (CCA) have an estimated prevalence ranging from 0.3% up to 0.7% in patients undergoing brain imaging. CCA can be identified incidentally, or can be part of a developmental disease. We performed a retrospective study of 551 patients, identified non-syndromic (NS) CCA and syndromic (S) CCA, reviewing clinical features, neuroradiological aspects, genetic etiology, and chromosomal microarray (CMA) results. Syndromic CCA subjects were prevalent (60%) and they showed the most severe clinical features. Cortical malformations and cerebellar anomalies were 23% of cerebral malformation associated to CCA (plus), 23 and 14% respectively in syndromic forms. A clinical and/or genetic diagnosis was obtained in 37% of syndromic CCA including chromosomal rearrangements on high-resolution karyotype (18%), microdeletion/microduplication syndromes (31%) and monogenic diseases (51%). Non-syndromic CCA anomalies had mildest clinical features, although intellectual disability was present in 49% of cases and epilepsy in 13%. CMA diagnostic rate in our cohort of patients ranged from 11 to 23% (NS to S). A high percentage of patients (76% 422/551) remain without a diagnosis. Combined high resolution CMA studies and next-generation sequencing (NGS) strategies will increase the probability to identify new causative genes of CCA and to redefine genotype–phenotype correlation