Bayesian model accounting for within-class biological variability in Serial Analysis of Gene Expression (SAGE)

Abstract Background An important challenge for transcript counting methods such as Serial Analysis of Gene Expression (SAGE), "Digital Northern" or Massively Parallel Signature Sequencing (MPSS), is to carry out statistical analyses that account for the within-class variability, i.e., variability due to the intrinsic biological differences among sampled individuals of the same class, and not only variability due to technical sampling error. Results We introduce a Bayesian model that accounts for the within-class variability by means of mixture distribution. We show that the previously available approaches of aggregation in pools ("pseudo-libraries") and the Beta-Binomial model, are particular cases of the mixture model. We illustrate our method with a brain tumor vs. normal comparison using SAGE data from public databases. We show examples of tags regarded as differentially expressed with high significance if the within-class variability is ignored, but clearly not so significant if one accounts for it. Conclusion Using available information about biological replicates, one can transform a list of candidate transcripts showing differential expression to a more reliable one. Our method is freely available, under GPL/GNU copyleft, through a user friendly web-based on-line tool or as R language scripts at supplemental web-site.

A theoretical model for estimating the margination constant of leukocytes

Abstract Background Blood leukocytes constitute two interchangeable sub-populations, the marginated and circulating pools. These two sub-compartments are found in normal conditions and are potentially affected by non-normal situations, either pathological or physiological. The dynamics between the compartments is governed by rate constants of margination (M) and return to circulation (R). Therefore, estimates of M and R may prove of great importance to a deeper understanding of many conditions. However, there has been a lack of formalism in order to approach such estimates. The few attempts to furnish an estimation of M and R neither rely on clearly stated models that precisely say which rate constant is under estimation nor recognize which factors may influence the estimation. Results The returning of the blood pools to a steady-state value after a perturbation (e.g., epinephrine injection) was modeled by a second-order differential equation. This equation has two eigenvalues, related to a fast- and to a slow-component of the dynamics. The model makes it possible to identify that these components are partitioned into three constants: R, M and SB; where SB is a time-invariant exit to tissues rate constant. Three examples of the computations are worked and a tentative estimation of R for mouse monocytes is presented. Conclusions This study establishes a firm theoretical basis for the estimation of the rate constants of the dynamics between the blood sub-compartments of white cells. It shows, for the first time, that the estimation must also take into account the exit to tissues rate constant, SB.

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

A systematic approach to model nonlinear systems using norm-bounded linear differential inclusions (NLDIs) is proposed in this paper. The resulting NLDI model is suitable for the application of linear control design techniques and, therefore, it is possible to fulfill certain specifications for the underlying nonlinear system, within an operating region of interest in the state-space, using a linear controller designed for this NLDI model. Hence, a procedure to design a dynamic output feedback controller for the NLDI model is also proposed in this paper. One of the main contributions of the proposed modeling and control approach is the use of the mean-value theorem to represent the nonlinear system by a linear parameter-varying model, which is then mapped into a polytopic linear differential inclusion (PLDI) within the region of interest. To avoid the combinatorial problem that is inherent of polytopic models for medium- and large-sized systems, the PLDI is transformed into an NLDI, and the whole process is carried out ensuring that all trajectories of the underlying nonlinear system are also trajectories of the resulting NLDI within the operating region of interest. Furthermore, it is also possible to choose a particular structure for the NLDI parameters to reduce the conservatism in the representation of the nonlinear system by the NLDI model, and this feature is also one important contribution of this paper. Once the NLDI representation of the nonlinear system is obtained, the paper proposes the application of a linear control design method to this representation. The design is based on quadratic Lyapunov functions and formulated as search problem over a set of bilinear matrix inequalities (BMIs), which is solved using a two-step separation procedure that maps the BMIs into a set of corresponding linear matrix inequalities. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.

Pair approximation for a model of vertical and horizontal transmission of parasites.

We apply Stochastic Dynamics method for a differential equations model, proposed by Marc Lipsitch and collaborators (Proc. R. Soc. Lond. B 260, 321, 1995), for which the transmission dynamics of parasites occurs from a parent to its offspring (vertical transmission), and by contact with infected host (horizontal transmission). Herpes, Hepatitis and AIDS are examples of diseases for which both horizontal and vertical transmission occur simultaneously during the virus spreading. Understanding the role of each type of transmission in the infection prevalence on a susceptible host population may provide some information about the factors that contribute for the eradication and/or control of those diseases. We present a pair mean-field approximation obtained from the master equation of the model. The pair approximation is formed by the differential equations of the susceptible and infected population densities and the differential equations of pairs that contribute to the former ones. In terms of the model parameters, we obtain the conditions that lead to the disease eradication, and set up the phase diagram based on the local stability analysis of fixed points. We also perform Monte Carlo simulations of the model on complete graphs and Erdös-Rényi graphs in order to investigate the influence of population size and neighborhood on the previous mean-field results; by this way, we also expect to evaluate the contribution of vertical and horizontal transmission on the elimination of parasite. Pair Approximation for a Model of Vertical and Horizontal Transmission of Parasites.

Numerical Model for the Dynamic Analysis of Pile Foundations

Programa de doctorado: Sistemas Inteligentes y Aplicaciones Numéricas en Ingeniería Instituto Universitario (SIANI)

3-D geometry reconstruction using a color image stereo pair and partial differential equations

[EN] In this work, we present a new model for a dense disparity estimation and the 3-D geometry reconstruction using a color image stereo pair. First, we present a brief introduction to the 3-D Geometry of a camera system. Next, we propose a new model for the disparity estimation based on an energy functional. We look for the local minima of the energy using the associate Euler-Langrage partial differential equations. This model is a generalization to color image of the model developed in, with some changes in the strategy to avoid the irrelevant local minima. We present some numerical experiences of 3-D reconstruction, using this method some real stereo pairs.

Dynamic analysis of piled foundations in stratified soils by a BEM-FEM model

[EN] 3D BEM-FEM coupling model is used to study the dynamic behavior of piled foundations in elastic layered soils in presenceof a rigid bedrock. Piles are modelled by FEM as beams according to the Bernoulli hpothesis, and every layer of the soil is modelled by BEM as a cointinuum, semi-infinite, isotropic, homogeneous, linear, viscoelastic medium.

BEM-FEM coupling model for the dynamic analysis of piles and pile groups

[EN] This paper shows a BEM-FEM coupling model for the time harmonic dynamic analysis of piles and pile groups embeddes in an elastic half-space. Piles are modelled using Finite Elements (FEM) as a beam according to the Bernoulli hypothesis, while the soil modelled using  Boundary Elements (BEM) as a continuum, semi-infinite, isotropic, homogeneous or zoned homogeneous, linear, viscoelastic medium.

Dynamic structure-soil-structure interaction between nearby piled buildings under seismic excitation by BEM-FEM model

[EN]The dynamic throug-soil interaction between nearby pile supported structures in a viscoelastic half-space, under incident S and Rayleigh waves, is numerically studied. To this end, a three-dimensional viscoelastic BEM-FEM formulation for the dynamic analysis of piles and pile groups in the frequency domain is used, where soil is modelled by BEM and piles are simulated by one-dimensional finite elements as Bernouilli beams.

Retrieval of trace gases vertical profile in the lower atmosphere combining. Differential Optical Absorption Spectroscopy with radiative transfer models

The motivation for the work presented in this thesis is to retrieve profile information for the atmospheric trace constituents nitrogen dioxide (NO2) and ozone (O3) in the lower troposphere from remote sensing measurements. The remote sensing technique used, referred to as Multiple AXis Differential Optical Absorption Spectroscopy (MAX-DOAS), is a recent technique that represents a significant advance on the well-established DOAS, especially for what it concerns the study of tropospheric trace consituents. NO2 is an important trace gas in the lower troposphere due to the fact that it is involved in the production of tropospheric ozone; ozone and nitrogen dioxide are key factors in determining the quality of air with consequences, for example, on human health and the growth of vegetation. To understand the NO2 and ozone chemistry in more detail not only the concentrations at ground but also the acquisition of the vertical distribution is necessary. In fact, the budget of nitrogen oxides and ozone in the atmosphere is determined both by local emissions and non-local chemical and dynamical processes (i.e. diffusion and transport at various scales) that greatly impact on their vertical and temporal distribution: thus a tool to resolve the vertical profile information is really important. Useful measurement techniques for atmospheric trace species should fulfill at least two main requirements. First, they must be sufficiently sensitive to detect the species under consideration at their ambient concentration levels. Second, they must be specific, which means that the results of the measurement of a particular species must be neither positively nor negatively influenced by any other trace species simultaneously present in the probed volume of air. Air monitoring by spectroscopic techniques has proven to be a very useful tool to fulfill these desirable requirements as well as a number of other important properties. During the last decades, many such instruments have been developed which are based on the absorption properties of the constituents in various regions of the electromagnetic spectrum, ranging from the far infrared to the ultraviolet. Among them, Differential Optical Absorption Spectroscopy (DOAS) has played an important role. DOAS is an established remote sensing technique for atmospheric trace gases probing, which identifies and quantifies the trace gases in the atmosphere taking advantage of their molecular absorption structures in the near UV and visible wavelengths of the electromagnetic spectrum (from 0.25 μm to 0.75 μm). Passive DOAS, in particular, can detect the presence of a trace gas in terms of its integrated concentration over the atmospheric path from the sun to the receiver (the so called slant column density). The receiver can be located at ground, as well as on board an aircraft or a satellite platform. Passive DOAS has, therefore, a flexible measurement configuration that allows multiple applications. The ability to properly interpret passive DOAS measurements of atmospheric constituents depends crucially on how well the optical path of light collected by the system is understood. This is because the final product of DOAS is the concentration of a particular species integrated along the path that radiation covers in the atmosphere. This path is not known a priori and can only be evaluated by Radiative Transfer Models (RTMs). These models are used to calculate the so called vertical column density of a given trace gas, which is obtained by dividing the measured slant column density to the so called air mass factor, which is used to quantify the enhancement of the light path length within the absorber layers. In the case of the standard DOAS set-up, in which radiation is collected along the vertical direction (zenith-sky DOAS), calculations of the air mass factor have been made using “simple” single scattering radiative transfer models. This configuration has its highest sensitivity in the stratosphere, in particular during twilight. This is the result of the large enhancement in stratospheric light path at dawn and dusk combined with a relatively short tropospheric path. In order to increase the sensitivity of the instrument towards tropospheric signals, measurements with the telescope pointing the horizon (offaxis DOAS) have to be performed. In this circumstances, the light path in the lower layers can become very long and necessitate the use of radiative transfer models including multiple scattering, the full treatment of atmospheric sphericity and refraction. In this thesis, a recent development in the well-established DOAS technique is described, referred to as Multiple AXis Differential Optical Absorption Spectroscopy (MAX-DOAS). The MAX-DOAS consists in the simultaneous use of several off-axis directions near the horizon: using this configuration, not only the sensitivity to tropospheric trace gases is greatly improved, but vertical profile information can also be retrieved by combining the simultaneous off-axis measurements with sophisticated RTM calculations and inversion techniques. In particular there is a need for a RTM which is capable of dealing with all the processes intervening along the light path, supporting all DOAS geometries used, and treating multiple scattering events with varying phase functions involved. To achieve these multiple goals a statistical approach based on the Monte Carlo technique should be used. A Monte Carlo RTM generates an ensemble of random photon paths between the light source and the detector, and uses these paths to reconstruct a remote sensing measurement. Within the present study, the Monte Carlo radiative transfer model PROMSAR (PROcessing of Multi-Scattered Atmospheric Radiation) has been developed and used to correctly interpret the slant column densities obtained from MAX-DOAS measurements. In order to derive the vertical concentration profile of a trace gas from its slant column measurement, the AMF is only one part in the quantitative retrieval process. One indispensable requirement is a robust approach to invert the measurements and obtain the unknown concentrations, the air mass factors being known. For this purpose, in the present thesis, we have used the Chahine relaxation method. Ground-based Multiple AXis DOAS, combined with appropriate radiative transfer models and inversion techniques, is a promising tool for atmospheric studies in the lower troposphere and boundary layer, including the retrieval of profile information with a good degree of vertical resolution. This thesis has presented an application of this powerful comprehensive tool for the study of a preserved natural Mediterranean area (the Castel Porziano Estate, located 20 km South-West of Rome) where pollution is transported from remote sources. Application of this tool in densely populated or industrial areas is beginning to look particularly fruitful and represents an important subject for future studies.

Jarrow-Yildirim model for inflation: theory and applications

This thesis deals with inflation theory, focussing on the model of Jarrow & Yildirim, which is nowadays used when pricing inflation derivatives. After recalling main results about short and forward interest rate models, the dynamics of the main components of the market are derived. Then the most important inflation-indexed derivatives are explained (zero coupon swap, year-on-year, cap and floor), and their pricing proceeding is shown step by step. Calibration is explained and performed with a common method and an heuristic and non standard one. The model is enriched with credit risk, too, which allows to take into account the possibility of bankrupt of the counterparty of a contract. In this context, the general method of pricing is derived, with the introduction of defaultable zero-coupon bonds, and the Monte Carlo method is treated in detailed and used to price a concrete example of contract. Appendixes: A: martingale measures, Girsanov's theorem and the change of numeraire. B: some aspects of the theory of Stochastic Differential Equations; in particular, the solution for linear EDSs, and the Feynman-Kac Theorem, which shows the connection between EDSs and Partial Differential Equations. C: some useful results about normal distribution.

A model for diffusive and displacive phase transitions in steels

Heat treatment of steels is a process of fundamental importance in tailoring the properties of a material to the desired application; developing a model able to describe such process would allow to predict the microstructure obtained from the treatment and the consequent mechanical properties of the material. A steel, during a heat treatment, can undergo two different kinds of phase transitions [p.t.]: diffusive (second order p.t.) and displacive (first order p.t.); in this thesis, an attempt to describe both in a thermodynamically consistent framework is made; a phase field, diffuse interface model accounting for the coupling between thermal, chemical and mechanical effects is developed, and a way to overcome the difficulties arising from the treatment of the non-local effects (gradient terms) is proposed. The governing equations are the balance of linear momentum equation, the Cahn-Hilliard equation and the balance of internal energy equation. The model is completed with a suitable description of the free energy, from which constitutive relations are drawn. The equations are then cast in a variational form and different numerical techniques are used to deal with the principal features of the model: time-dependency, non-linearity and presence of high order spatial derivatives. Simulations are performed using DOLFIN, a C++ library for the automated solution of partial differential equations by means of the finite element method; results are shown for different test-cases. The analysis is reduced to a two dimensional setting, which is simpler than a three dimensional one, but still meaningful.

Use of VEGFR-2 targeted microbubbles (BR55, Bracco imaging) for the early ultrasound evaluation of response to antiangiogenic treatment in a xenograft model of hepatocarcinoma

Aim: To evaluate the early response to treatment to an antiangiogenetic drug (sorafenib) in a heterotopic murine model of hepatocellular carcinoma (HCC) using ultrasonographic molecular imaging. Material and Methods: the xenographt model was established injecting a suspension of HuH7 cells subcutaneously in 19 nude mice. When tumors reached a mean diameter of 5-10 mm, they were divided in two groups (treatment and vehicle). The treatment group received sorafenib (62 mg/kg) by daily oral gavage for 14 days. Molecular imaging was performed using contrast enhanced ultrasound (CEUS), by injecting into the mouse venous circulation a suspension of VEGFR-2 targeted microbubbles (BR55, kind gift of Bracco Swiss, Geneve, Switzerland). Video clips were acquired for 6 minutes, then microbubbles (MBs) were destroyed by a high mechanical index (MI) impulse, and another minute was recorded to evaluate residual circulating MBs. The US protocol was repeated at day 0,+2,+4,+7, and +14 from the beginning of treatment administration. Video clips were analyzed using a dedicated software (Sonotumor, Bracco Swiss) to quantify the signal of the contrast agent. Time/intensity curves were obtained and the difference of the mean MBs signal before and after high MI impulse (Differential Targeted Enhancement-dTE) was calculated. dTE represents a numeric value in arbitrary units proportional to the amount of bound MBs. At day +14 mice were euthanized and the tumors analyzed for VEGFR-2, pERK, and CD31 tissue levels using western blot analysis. Results: dTE values decreased from day 0 to day +14 both in treatment and vehicle groups, and they were statistically higher in vehicle group than in treatment group at day +2, at day +7, and at day +14. With respect to the degree of tumor volume increase, measured as growth percentage delta (GPD), treatment group was divided in two sub-groups, non-responders (GPD>350%), and responders (GPD<200%). In the same way vehicle group was divided in slow growth group (GPD<400%), and fast growth group (GPD>900%). dTE values at day 0 (immediately before treatment start) were higher in non-responders than in responders group, with statistical difference at day 2. While dTE values were higher in the fast growth group than in the slow growth group only at day 0. A significant positive correlation was found between VEGFR-2 tissue levels and dTE values, confirming that level of BR55 tissue enhancement reflects the amount of tissue VEGF receptor. Conclusions: the present findings show that, at least in murine experimental models, CEUS with BR55 is feasable and appears to be a useful tool in the prediction of tumor growth and response to sorafenib treatment in xenograft HCC.

Differential cell type-specific transcriptional regulation of the CYP1A1 gene

Cytochrome P450 1A1 (CYP1A1) monooxygenase plays an important role in the metabolism of environmental pollutants such as polycyclic aromatic hydrocarbons (PAHs) and halogenated polycyclic aromatic hydrocarbons (HAHs). Oxidation of these compounds converts them to the metabolites that subsequently can be conjugated to hydrophilic endogenous entities e.g. glutathione. Derivates generated in this way are water soluble and can be excreted in bile or urine, which is a defense mechanism. Besides detoxification, metabolism by CYP1A1 may lead to deleterious effects since the highly reactive intermediate metabolites are able to react with DNA and thus cause mutagenic effects, as it is in the case of benzo(a) pyrene (B[a]P). CYP1A1 is normally not expressed or expressed at a very low level in the cells but it is inducible by many PAHs and HAHs e.g. by B[a]P or 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD). Transcriptional activation of the CYP1A1 gene is mediated by aryl hydrocarbon receptor (AHR), a basic-helix-loop-helix (bHLH) transcription factor. In the absence of a ligand AHR stays predominantly in the cytoplasm. Ligand binding causes translocation of AHR to the nuclear compartment, its heterodimerization with another bHLH protein, the aryl hydrocarbon nuclear translocator (ARNT) and binding of the AHR/ARNT heterodimer to a DNA motif designated dioxin responsive element (DRE). This process leads to the transcriptional activation of the responsive genes containing DREs in their regulatory regions, e.g. that coding for CYP1A1. TCDD is the most potent known agonist of AHR. Since it is not metabolized by the activated enzymes, exposure to this compound leads to a persisting activation of AHR resulting in diverse toxic effects in the organism. To enlighten the molecular mechanisms that mediate the toxicity of xenobiotics like TCDD and related compounds, the AHR-dependent regulation of the CYP1A1 gene was investigated in two cell lines: human cervix carcinoma (HeLa) and mouse hepatoma (Hepa). Study of AHR activation and its consequence concerning expression of the CYP1A1 enzyme confirmed the TCDD-dependent formation of the AHR/ARNT complex on DRE leading to an increase of the CYP1A1 transcription in Hepa cells. In contrast, in HeLa cells formation of the AHR/ARNT heterodimer and binding of a protein complex containing AHR and ARNT to DRE occurred naturally in the absence of TCDD. Moreover, treatment with TCDD did not affect the AHR/ARNT dimer formation and binding of these proteins to DRE in these cells. Even though the constitutive complex on DRE exists in HeLa, transcription of the CYP1A1 gene was not increased. Furthermore, the CYP1A1 level in HeLa cells remained unchanged in the presence of TCDD suggesting repressional mechanism of the AHR complex function which may hinder the TCDD-dependent mechanisms in these cells. Similar to the native, the mouse CYP1A1-driven reporter constructs containing different regulatory elements were not inducible by TCDD in HeLa cells, which supported a presence of cell type specific trans-acting factor in HeLa cells able to repress both the native CYP1A1 and CYP1A1-driven reporter genes rather than species specific differences between CYP1A1 genes of human and rodent origin. The different regulation of the AHR-mediated transcription of CYP1A1 gene in Hepa and HeLa cells was further explored in order to elucidate two aspects of the AHR function: (I) mechanism involved in the activation of AHR in the absence of exogenous ligand and (II) factor that repress function of the exogenous ligand-independent AHR/ARNT complex. Since preliminary studies revealed that the activation of PKA causes an activation of AHR in Hepa cells in the absence of TCDD, the PKA-dependent signalling pathway was the proposed endogenous mechanism leading to the TCDD-independent activation of AHR in HeLa cells. Activation of PKA by forskolin or db-cAMP as well as inhibition of the kinase by H89 in both HeLa and Hepa cells did not lead to alterations in the AHR interaction with ARNT in the absence of TCDD and had no effect on binding of these proteins to DRE. Moreover, the modulators of PKA did not influence the CYP1A1 activity in these cells in the presence and in the absence of TCDD. Thus, an involvement of PKA in the regulation of the CYP1A1 Gen in HeLa cells was not evaluated in the course of this study. Repression of genes by transcription factors bound to their responsive elements in the absence of ligands has been described for nuclear receptors. These receptors interact with protein complex containing histone deacetylase (HDAC), enzyme responsible for the repressional effect. Thus, a participation of histone deacetylase in the transcriptional modulation of CYP1A1 gene by the constitutively DNA-bound AHR/ARNT complex was supposed. Inhibition of the HDAC activity by trichostatin A (TSA) or sodium butyrate (NaBu) led to an increase of the CYP1A1 transcription in the presence but not in the absence of TCDD in Hepa and HeLa cells. Since amount of the AHR and ARNT proteins remained unchanged upon treatment of the cells with TSA or NaBu, the transcriptional upregulation of CYP1A1 gene was not due to an increased expression of the regulatory proteins. These findings strongly suggest an involvement of HDAC in the repression of the CYP1A1 gene. Similar to the native human CYP1A1 also the mouse CYP1A1-driven reporter gene transfected into HeLa cells was repressed by histone deacetylase since the presence of TSA or NaBu led to an increase in the reporter activity. Induction of reporter gene did not require a presence of the promoter or negative regulatory regions of the CYP1A1 gene. A promoter-distal fragment containing three DREs together with surrounding sequences was sufficient to mediate the effects of the HDAC inhibitors suggesting that the AHR/ARNT binding to its specific DNA recognition site may be important for the CYP1A1 repression. Histone deacetylase is recruited to the specific genes by corepressors, proteins that bind to the transcription factors and interact with other members of the HDAC complex. Western blot analyses revealed a presence of HDAC1 and the corepressors mSin3A (mammalian homolog of yeast Sin3) and SMRT (silencing mediator for retinoid and thyroid hormone receptor) in both cell types, while the corepressor NCoR (nuclear receptor corepressor) was expressed exclusively in HeLa cells. Thus the high inducibility of CYP1A1 in Hepa cells may be due to the absence of NCoR in these cells in contrast to the non-responsive HeLa cells, where the presence of NCoR would support repression of the gene by histone deacetylase. This hypothesis was verified in reporter gene experiments where expression constructs coding for the particular members of the HDAC complex were cotransfected in Hepa cells together with the TCDD-inducible reporter constructs containing the CYP1A1 regulatory sequences. An overexpression of NCoR however did not decrease but instead led to a slight increase of the reporter gene activity in the cells. The expected inhibition was observed solely in the case of SMRT that slightly reduced constitutive and TCDD-induced reporter gene activity. A simultaneous expression of NCoR and SMRT shown no further effects and coexpression of HDAC1 with the two corepressors did not alter this situation. Thus, additional factors that are likely involved in the repression of CYP1A1 gene by HDAC complex remained to be identified. Taking together, characterisation of an exogenous ligand independent AHR/ARNT complex on DRE in HeLa cells that repress transcription of the CYP1A1 gene creates a model system enabling investigation of endogenous processes involved in the regulation of AHR function. This study implicates HDAC-mediated repression of CYP1A1 gene that contributes to the xenobiotic-induced expression in a tissue specific manner. Elucidation of these processes gains an insight into mechanisms leading to deleterious effects of TCDD and related compounds.

Classical limit of the Nelson model

Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.