Andrea Signori

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• Ph.D. in Mathematics
• Rtda at Politecnico di Milano

This paper is intended to tackle the control problem associated with an extended phase field system of Cahn–Hilliard type that is related to a tumor growth model. This system has been investigated in previous contributions from the viewpoint of well-posedness and asymptotic analyses. Here, we aim to extend the mathematical studies around this syste...

The Cahn--Hilliard equation is one of the most common models to describe phase segregation processes in binary mixtures. In recent times, various dynamic boundary conditions have been introduced to model interactions of the materials with the boundary more precisely. To take long-range interactions of the materials into account, we propose a new mo...

Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn-Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell-cell adhesion effects are taken into account with the help of a Ginzburg--Landau type energy. In the overall model an equation of Cahn-Hil...

We propose a new type of diffuse interface model describing the evolution of a tumor mass under the effects of a chemical substance (e.g., a nutrient or a drug). The process is described by utilizing the variables φ, an order parameter representing the local proportion of tumor cells, and σ, representing the concentration of the chemical. The order...

In the present work, we develop a comprehensive and rigorous analytical framework for a non-local phase-field model that describes tumour growth dynamics. The model is derived by coupling a non-local Cahn-Hilliard equation with a parabolic reaction-diffusion equation, which accounts for both phase segregation and nutrient diffusion. Previous studie...

The Cahn-Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn-Hilliard equation has been formulated...

This paper presents an existence result for the anisotropic Cahn--Hilliard equation characterized by a potentially concentration-dependent degenerate mobility taking into account an anisotropic energy. The model allows for the degeneracy of the mobility at specific concentration values, demonstrating that the solution remains within physically rele...

We investigate a new diffuse-interface model that describes creeping two-phase flows (i.e., flows exhibiting a low Reynolds number), especially flows that permeate a porous medium. The system of equations consists of a Brinkman equation for the volume averaged velocity field and a convective Cahn–Hilliard equation with dynamic boundary conditions f...

This paper investigates a Cahn–Hilliard–Swift–Hohenberg system, focusing on a three-species chemical mixture subject to physical constraints on volume fractions. The resulting system leads to complex patterns involving a separation into phases as typical of the Cahn–Hilliard equation and small scale stripes and dots as seen in the Swift–Hohenberg e...

This paper investigates an optimal control problem associated with a two-dimensional multi-species Cahn-Hilliard-Keller-Segel tumor growth model, which incorporates complex biological processes such as species diffusion, chemotaxis, angiogenesis, and nutrient consumption, resulting in a highly nonlinear system of nonlinear partial differential equa...

In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn–Hilliard equation endowed with the singular Flory-Huggins potential in the three-dimensional setting. The control enters the governing state system in a nonlinear fashion in the form of a prescribed solenoidal, that is a divergence-free, vector field, whereas...

In this note, we study the optimal control of a nonisothermal phase field system of Cahn–Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn–Hilliard type equation with source term for the order parameter with the universal balanc...

We analyze a phase field model for tumor growth consisting of a Cahn–Hilliard–Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all...

In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn-Hilliard equation endowed with the singular Flory-Huggins potential in the three-dimensional setting. The control enters the governing state system in a nonlinear fashion in the form of a prescribed solenoidal, that is a divergence-free, vector field, whereas...

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of...

A phase field approach for structural topology optimization with application to additive manufacturing is analyzed. The main novelty is the penalization of overhangs (regions of the design that require underlying support structures during construction) with anisotropic energy functionals. Convex and non-convex examples are provided, with the latter...

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balanc...

This work concerns a structural topology optimisation problem for 4D printing based on the phase field approach. The concept of 4D printing as a targeted evolution of 3D printed structures can be realised in a two-step process. One first fabricates a 3D object with multi-material active composites and apply external loads in the programming stage....

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of...

A nonisothermal phase field system of Cahn-Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law...

This work concerns a structural topology optimisation problem for 4D printing based on the phase field approach. The concept of 4D printing as a targeted evolution of 3D printed structures can be realised in a two-step process. One first fabricates a 3D object with multi-material active composites and apply external loads in the programming stage....

This paper treats a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermo...

In this paper, we investigate optimal control problems for a nonlinear state system which constitutes a version of the Caginalp phase field system modeling nonisothermal phase transitions with a nonconserved order parameter that takes thermal memory into account. The state system, which is a first-order approximation of a thermodynamically consiste...

We analyze a phase field model for tumor growth consisting of a Cahn-Hilliard-Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all...

A phase field model which describes the formation of protein-RNA complexes subject to phase segregation is analyzed. A single protein, two RNA species, and two complexes are considered. Protein and RNA species are governed by coupled reaction-diffusion equations which also depend on the two complexes. The latter ones are driven by two Cahn-Hilliard...

We propose a new type of diffuse interface model describing the evolution of a tumor mass under the effects of a chemical substance (e.g., a nutrient or a drug). The process is described by utilizing the variables $\varphi$, an order parameter representing the local proportion of tumor cells, and $\sigma$, representing the concentration of the chem...

A Correction to this paper has been published: https://doi.org/10.1007/s00245-019-09618-6

A phase field approach for structural topology optimization with application to additive manufacturing is analyzed. The main novelty is the penalization of overhangs (regions of the design that require underlying support structures during construction) with anisotropic energy functionals. Convex and non-convex examples are provided, with the latter...

In this paper, we introduce the problem of parameter identification for a coupled nonlocal Cahn–Hilliard-reaction-diffusion PDE system stemming from a recently introduced tumor growth model. The inverse problem of identifying relevant parameters is studied here by relying on techniques from optimal control theory of PDE systems. The parameters to b...

We investigate a multiphase Cahn–Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stra...

A nonlinear extension of the Caginalp phase field system is considered that takes thermal memory into account. The resulting model, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. Two equations, resulting from phase dynamics and the universal balance law for i...

This paper concerns a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type including chemotaxis with possibly singular potentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singul...

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential s...

We investigate a multiphase Cahn-Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stra...

This paper provides a unified mathematical analysis of a family of non-local diffuse interface models for tumor growth describing evolutions driven by long-range interactions. These integro-partial differential equations model cell-to-cell adhesion by a non-local term and may be seen as non-local variants of the corresponding local model proposed b...

This paper treats a distributed optimal control problem for a tumor growth model of Cahn-Hilliard type including chemotaxis. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly c...

In this paper, we study an optimal control problem for a macroscopic mechanical tumor model based on the phase field approach. The model couples a Cahn-Hilliard-type equation to a system of linear elasticity and a reaction-diffusion equation for a nutrient concentration. By taking advantage of previous analytical well-posedness results established...

The Cahn–Hilliard equation is one of the most common models to describe phase segregation processes in binary mixtures. Various dynamic boundary conditions have already been introduced in the literature to model interactions of the materials with the boundary more precisely. To take long-range interactions into account, we propose a new model consi...

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs , Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng. 28...

Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hill...

A distributed optimal control problem for a diffuse interface model, which physical context is that of tumour growth dynamics, is addressed. The system we deal with comprises a Cahn-Hilliard equation for the tumour fraction coupled with a reaction-diffusion for a nutrient species surrounding the tumourous cells. The cost functional to be minimised...

In this paper, we study an optimal control problem for a macroscopic mechanical tumour model based on the phase field approach. The model couples a Cahn--Hilliard type equation to a system of linear elasticity and a reaction-diffusion equation for a nutrient concentration. By taking advantage of previous analytical well-posedness results establishe...

We introduce the problem of parameter identification for a coupled nonlocal Cahn-Hilliard-reaction-diffusion PDE system stemming from a recently introduced tumor growth model. The inverse problem of identifying relevant parameters is studied here by relying on techniques from optimal control theory of PDE systems. The parameters to be identified pl...

This paper concerns a distributed optimal control problem for a tumor growth model of Cahn-Hilliard type including chemotaxis with possibly singular potentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singul...

A distributed optimal control problem for a diffuse interface model, which physical context is that of tumour growth dynamics, is addressed. The system we deal with comprises a Cahn-Hilliard equation for the tumour fraction coupled with a reaction-diffusion for a nutrient species surrounding the tumourous cells. The cost functional to be minimise...

We extend previous weak well-posedness results obtained in Frigeri et al. (2017) concerning a non-local variant of a diffuse interface tumor model proposed by Hawkins-Daarud et al. (2012). The model consists of a non-local Cahn--Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction-dif...

This paper provides a unified mathematical analysis of a family of non-local diffuse interface models for tumor growth. These are non-local variants of the corresponding local model proposed by H. Garcke et al. (2016), and take into account the long-range interactions occurring in biological phenomena. The model in consideration couples a nonlocal...

This paper is concerned with a boundary control problem for the Cahn–Hilliard equation coupled with dynamic boundary conditions. In order to handle the control problem, we restrict our analysis to the case of regular potentials defined on the whole real line, assuming the boundary potential to be dominant. The existence of optimal control, the Fréc...

A distributed optimal control problem for a phase field system which physical context is that of tumor growth is discussed. The system we are going to take into account consists of a Cahn-Hilliard equation for the phase variable (relative concentration of the tumor), coupled with a reaction-diffusion equation for the nutrient. The cost functional i...

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential s...

This paper is concerned with a boundary control problem for the Cahn--Hilliard equation coupled with dynamic boundary conditions. In order to handle the control problem, we restrict our analysis to the case of regular potentials defined on the whole real line, assuming the boundary potential to be dominant. The existence of optimal control, the Fr\...

This paper is concerned with a boundary control problem for the Cahn-Hilliard equation coupled with dynamic boundary conditions. In order to handle the control problem, we restrict our analysis to the case of regular potentials defined on the whole real line, assuming the boundary potential to be dominant. The existence of optimal control, the Fréc...

A distributed optimal control problem for a phase field system which physical context is that of tumor growth is discussed. The system we are going to take into account consists of a Cahn-Hilliard equation for the phase variable (relative concentration of the tumor), coupled with a reaction-diffusion equation for the nutrient. The cost functional i...

We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard type equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction-diffusion equation for the nutrient. This mod...

We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction diffusion equation for the nutrient. This model ha...

In this work, we investigate a distributed optimal control problem for an extended phase field system of Cahn--Hilliard type which physical context is that of tumor growth dynamics. In a previous contribution, the author has already studied the corresponding problem for the logarithmic potential. Here, we try to extend the analysis by taking into a...

In this work, we investigate a distributed optimal control problem for an extended phase field system of Cahn-Hilliard type which physical context is that of tumor growth dynamics. In a previous contribution, the author has already studied the corresponding problem for the logarithmic potential. Here, we try to extend the analysis by taking into ac...

This paper is intended to tackle the control problem associated with an extended phase field system of Cahn-Hilliard type that is related to a tumor growth model. This system has been investigated in previous contributions from the viewpoint of well-posedness and asymptotic analyses. Here, we aim to extend the mathematical studies around this syste...