arXiv subjects

Elisabetta Rocca

Publications and source records attributed to Elisabetta Rocca.

At least 37 records · Page 2Linked to original sources

arXiv2023

Strict separation and numerical approximation for a non-local Cahn-Hilliard equation with single-well potential

In this paper we study a non-local Cahn-Hilliard equation with singular single-well potential and degenerate mobility. This results as a particular case of a more general model derived for a binary, saturated, closed and incompressible mixture, composed by a tumor phase and a healthy phase, evolving in a bounded domain. The general system couples a Darcy-type evolution for the average velocity field with a convective reaction-diffusion type evolution for the nutrient concentration and a non-local convective Cahn-Hilliard equation for the tumor phase. The main mathematical difficulties are related to the proof of the separation property for the tumor phase in the Cahn-Hilliard equation: up to our knowledge, such problem is indeed open in the literature. For this reason, in the present contribution we restrict the analytical study to the Cahn-Hilliard equation only. For the non-local Cahn- Hilliard equation with singular single-well potential and degenerate mobility, we study the existence and uniqueness of weak solutions for spatial dimensions $d\leq 3$. After showing existence, we prove the strict separation property in three spatial dimensions, implying the same property also for lower spatial dimensions, which opens the way to the proof of uniqueness of solutions. Finally, we propose a well posed and gradient stable continuous finite element approximation of the model for $d\leq 3$, which preserves the physical properties of the continuos solution and which is computationally efficient, and we show simulation results in two spatial dimensions which prove the consistency of the proposed scheme and which describe the phase ordering dynamics associated to the system.

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arXiv2023

Optimal control of a reaction-diffusion model related to the spread of COVID-19

This paper is concerned with the well-posedness and optimal control problem of a reaction-diffusion system for an epidemic Susceptible-Infected-Recovered-Susceptible (SIRS) mathematical model in which the dynamics develops in a spatially heterogeneous environment. Using as control variables the transmission rates $u_{i}$ and $u_{e}$ of contagion resulting from the contact with both asymptomatic and symptomatic persons, respectively, we optimize the number of exposed and infected individuals at a final time $T$ of the controlled evolution of the system. More precisely, we search for the optimal $u_{i}$ and $u_{e}$ such that the number of infected plus exposed does not exceed at the final time a threshold value $Λ$, fixed a priori. We prove here the existence of optimal controls in a proper functional framework and we derive the first-order necessary optimality conditions in terms of the adjoint variables.

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arXiv2023

Well-posedness for a diffusion-reaction compartmental model simulating the spread of COVID-19

This paper is concerned with the well-posedness of a diffusion-reaction system for a Susceptible-Exposed-Infected-Recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio-temporal spread of the COVID-19 pandemic and is a variation of the one recently introduced, discussed and tested in [A. Viguerie et al, Diffusion-reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study, Comput. Mech. 66 (2020) 1131-1152]. Here, we deal with the mathematical analysis of the resulting Cauchy-Neumann problem: the existence of solutions is proved in a rather general setting and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions.

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arXiv2023

A Cahn-Hilliard phase field model coupled to an Allen-Cahn model of viscoelasticity at large strains

We propose a new Cahn-Hilliard phase field model coupled to incompressible viscoelasticity at large strains, obtained from a diffuse interface mixture model and formulated in the Eulerian configuration. A new kind of diffusive regularization, of Allen-Cahn type, is introduced in the transport equation for the deformation gradient, together with a regularizing interface term depending on the gradient of the deformation gradient in the free energy of the system. We study the global existence of a weak solution for the model. While standard diffusive regularizations of the transport equation for the deformation gradient presented in literature allows the existence study only for simplified cases, i.e. in two space dimensions and for convex elastic free energy densities of Neo-Hookean type which are independent from the phase field variable, the present regularization allows to study more general cases. In particular, we obtain the global existence of a weak solution in three space dimensions and for generic nonlinear elastic energy densities with polynomial growth. Our analysis considers elastic free energy densities which depend on the phase field variable and which can possibly degenerate for some values of the phase field variable. By means of an iterative argument based on elliptic regularity bootstrap steps, we find the maximum allowed polynomial growths of the Cahn-Hilliard potential and the elastic energy density which guarantee the existence of a solution in three space dimensions. We propose two unconditionally energy stable finite element approximations of the model, based on convex splitting ideas and on the use of a scalar auxiliary variable, proving the existence and stability of discrete solutions. We finally report numerical results for different test cases with shape memory alloy type free energy with pure phases characterized by different elastic properties.

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arXiv2022

Identification of cavities and inclusions in linear elasticity with a phase-field approach

In this paper, we deal with the inverse problem of the shape reconstruction of cavities and inclusions embedded in a linear elastic isotropic medium from boundary displacement's measurements. For, we consider a constrained minimization problem involving a boundary quadratic misfit functional with a regularization term that penalizes the perimeter of the cavity or inclusion to be identified. Then using a phase-field approach we derive a robust algorithm for the reconstruction of elastic inclusions and of cavities modeled as inclusions with a very small elasticity tensor.

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arXiv2022

Well-posedness and optimal control for a Cahn-Hilliard-Oono system with control in the mass term

The paper treats the problem of optimal distributed control of a Cahn-Hilliard-Oono system in $\mathbb{R}^d$, $1\leq d\leq 3$, with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. In the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case $d = 2$. In the rest of the work, we study the necessary first-order optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain. PLEASE NOTE: A revised version of this paper has been published in Discrete Contin. Dyn. Syst. Ser. S 15 (2022), 2135-2172: the authors point out that both the published version and the arXiv preprint contain a trivial mistake in formula (2.21), which leads to suitable changes in the assumption (2.10) and in some occurrences of it. Namely, (2.21) and (2.10) become $ \bar φ(t) = \bar φ_0 e^{-t} + \int_0^t e^{-(t-s)} \bar u(s) \, ds$ for every $t\in[0,T]$, and $-M - (\barφ_0)^- , M + (\barφ_0)^+$ belong to the interior of $D(β)$, respectively. The formula just after (2.21) becomes $-M - (\barφ_0)^- \leq \bar φ(t) \leq M + (\barφ_0)^+$ and a similar change must be done in (4.20). The first line of (2.24) is now implied by the new (2.21). Next, (2.39) becomes $-(M+ R) - (\barφ_0)^- , (M+ R) + (\barφ_0)^+$ belong to the interior of $D(β)$. Similar changes must be done in the interval written just two lines before (4.18) and in (4.18) itself.

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arXiv2022

A Cahn-Hilliard model coupled to viscoelasticity with large deformations

We propose a new class of phase field models coupled to viscoelasticity with large deformations, obtained from a diffuse interface mixture model composed by a phase with elastic properties and a liquid phase. The model is formulated in the Eulerian configuration and it is derived by imposing the mass balance for the mixture components and the momentum balance that comes from a generalized form of the principle of virtual powers. The latter considers the presence of a system of microforces and microstresses associated to the microscopic interactions between the mixture's constituents together with a system of macroforces and macrostresses associated to their viscoelastic behavior, taking into account also the friction between the phases. The free energy density of the system is given as the sum of a Cahn-Hilliard term and an elastic polyconvex term, with a coupling between the phase field variable and the elastic deformation gradient in the elastic contribution. General constitutive assumptions complying with a mechanical version of the second law of thermodynamics in isothermal situations are taken. We study the global existence of a weak solution for a simplified and regularized version of the general model, which considers an incompressible elastic free energy of Neo-Hookean type with elastic coefficients depending on the phase field variable. The regularization is properly designed to deal with the coupling between the phase field variable and the elastic deformation gradient in the elastic energy density. The analysis is made both in two and three space dimensions.

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arXiv2022

On a Cahn-Hilliard-Keller-Segel model with generalized logistic source describing tumor growth

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 parameter $φ$ is assumed to satisfy a suitable form of the Cahn-Hilliard equation with mass source and logarithmic potential of Flory-Huggins type (or generalizations of it). The chemical concentration $σ$ satisfies a reaction-diffusion equation where the cross-diffusion term has the same expression as in the celebrated Keller-Segel model. In this respect, the model we propose represents a new coupling between the Cahn-Hilliard equation and a subsystem of the Keller-Segel model. We believe that, compared to other models, this choice is more effective in capturing the chemotactic effects that may occur in tumor growth dynamics (chemically induced tumor evolution and consumption of nutrient/drug by tumor cells). Note that, in order to prevent finite time blowup of $σ$, we assume a chemical source term of logistic type. Our main mathematical result is devoted to proving existence of weak solutions in a rather general setting that covers both the two- and three- dimensional cases. Under more restrictive assumptions on coefficients and data, and in some cases on the spatial dimension, we prove various regularity results. Finally, in a proper class of smooth solutions we show uniqueness and continuous dependence on the initial data in a number of significant cases.

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arXiv2021

Analysis of a tumor model as a multicomponent deformable porous medium

We propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn--Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces.

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arXiv2021

Mixed variational formulations for structural topology optimization based on the phase-field approach

We propose a variational principle combining a phase-field functional for structural topology optimization with a mixed (three-field) Hu-Washizu functional, then including directly in the formulation equilibrium, constitutive, and compatibility equations. The resulting mixed variational functional is then specialized to derive a classical topology optimization formulation (where the amount of material to be distributed is an \emph{a priori} assigned quantity acting as a global constraint for the problem) as well as a novel topology optimization formulation (where the amount of material to be distributed is minimized, hence with no pre imposed constraint for the problem). Both formulations are numerically solved by implementing a mixed finite element scheme, with the second approach avoiding the introduction of a global constraint, hence respecting the convenient local nature of the finite element discretization. Furthermore, within the proposed approach it is possible to obtain guidelines for settings proper values of phase-field-related simulation parameters and, thanks to the combined phase-field and Hu-Washizu rationale, a monolithic algorithm solution scheme can be easily adopted. An insightful and extensive numerical investigation results in a detailed convergence study and a discussion on the obtained final designs. The numerical results clearly highlight differences between the two formulations as well as advantages related to the monolithic solution strategy; numerical investigations address both two-dimensional and three-dimensional applications.

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arXiv2020

Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects

We consider a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects which is introduced in [2]. It is comprised of phase-field equation to describe tumor growth, which is coupled to a reaction-diffusion type equation for generic nutrient for the tumor. An additional equation couples the concentration of prostate-specific antigen (PSA) in the prostatic tissue and it obeys a linear reaction-diffusion equation. The system completes with homogeneous Dirichlet boundary conditions for the tumor variable and Neuman boundary condition for the nutrient and the concentration of PSA. Here we investigate the long time dynamics of the model. We first prove that the initial-boundary value problem generates a strongly continuous semigroup on a suitable phase space that admits the global attractor in a proper phase space. Moreover, we also discuss the convergence of a solution to a single stationary state and obtain a convergence rate estimate under some conditions on the coefficients.

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arXiv2020

On the Existence of Strong Solutions to the Cahn-Hilliard-Darcy system with mass source

We study a diffuse interface model describing the evolution of the flow of a binary fluid in a Hele-Shaw cell. The model consists of a Cahn-Hilliard-Darcy (CHD) type system with transport and mass source. A relevant physical application is related to tumor growth dynamics, which in particular justifies the occurrence of a mass inflow. We study the initial-boundary value problem for this model and prove global existence and uniqueness of strong solutions in two space dimensions as well as local existence in three space dimensions.

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arXiv2020

Parameter identification for nonlocal phase field models for tumor growth via optimal control and asymptotic analysis

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 play the role of controls, and a suitable cost functional of tracking-type is introduced to account for the discrepancy between some a priori knowledge of the parameters and the controls themselves. The analysis is carried out for several classes of models, each one depending on a specific relaxation (of parabolic or viscous type) performed on the original system. First-order necessary optimality conditions are obtained on the fully relaxed system, in both the two and three-dimensional case. Then, the optimal control problem on the non-relaxed models is tackled by means of asymptotic arguments, by showing convergence of the respective adjoint systems and the minimization problems as each one of the relaxing coefficients vanishes. This allows obtaining the desired necessary optimality conditions, hence to solve the parameter identification problem, for the original PDE system in case of physically relevant double-well potentials.

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arXiv2020

Optimal control of cytotoxic and antiangiogenic therapies on prostate cancer growth

Prostate cancer can be lethal in advanced stages, for which chemotherapy may become the only viable therapeutic option. While there is no clear clinical management strategy fitting all patients, cytotoxic chemotherapy with docetaxel is currently regarded as the gold standard. However, tumors may regain activity after treatment conclusion and become resistant to docetaxel. This situation calls for new delivery strategies and drug compounds enabling an improved therapeutic outcome. Combination of docetaxel with antiangiogenic therapy has been considered a promising strategy. Bevacizumab is the most common antiangiogenic drug, but clinical studies have not revealed a clear benefit from its combination with docetaxel. Here, we capitalize on our prior work on mathematical modeling of prostate cancer growth subjected to combined cytotoxic and antiangiogenic therapies, and propose an optimal control framework to robustly compute the drug-independent cytotoxic and antiangiogenic effects enabling an optimal therapeutic control of tumor dynamics. We describe the formulation of the optimal control problem, for which we prove the existence of at least a solution and determine the necessary first order optimality conditions. We then present numerical algorithms based on isogeometric analysis to run a preliminary simulation study over a single cycle of combined therapy. Our results suggest that only cytotoxic chemotherapy is required to optimize therapeutic performance and we show that our framework can produce superior solutions to combined therapy with docetaxel and bevacizumab. We also illustrate how the optimal drug-naïve cytotoxic effects computed in these simulations may be successfully leveraged to guide drug production and delivery strategies by running a nonlinear least-square fit of protocols involving docetaxel and a new design drug.

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arXiv2019

Optimal control of stochastic phase-field models related to tumor growth

We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.

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arXiv2019

Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model

In this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.

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arXiv2019

Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects

Cytotoxic chemotherapy is a common treatment for advanced prostate cancer. These tumors are also known to rely on angiogenesis, i.e., the growth of local microvasculature via chemical signaling produced by the tumor. Thus, several clinical studies have been investigating antiangiogenic therapy for advanced prostate cancer, either as monotherapy or combined with standard cytotoxic protocols. However, the complex genetic alterations promoting prostate cancer growth complicate the selection of the best chemotherapeutic approach for each patient's tumor. Here, we present a mathematical model of prostate cancer growth and chemotherapy that may enable physicians to test and design personalized chemotherapeutic protocols in silico. We use the phase-field method to describe tumor growth, which we assume to be driven by a generic nutrient following reaction-diffusion dynamics. Tumor proliferation and apoptosis (i.e., programmed cell death) can be parameterized with experimentally-determined values. Cytotoxic chemotherapy is included as a term downregulating tumor net proliferation, while antiangiogenic therapy is modeled as a reduction in intratumoral nutrient supply. Another equation couples the tumor phase field with the production of prostate-specific antigen, which is an extensively used prostate cancer biomarker. We prove the well-posedness of our model and we run a series of representative simulations using an isogeometric method to explore untreated tumor growth as well as the effects of cytotoxic chemotherapy and antiangiogenic therapy, both alone and combined. Our simulations show that our model captures the growth morphologies of prostate cancer as well as common outcomes of cytotoxic and antiangiogenic mono and combined therapy. Our model also reproduces the usual temporal trends in tumor volume and prostate-specific antigen evolution observed in previous studies.

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arXiv2019

Structural multiscale topology optimization with stress constraint for additive manufacturing

In this paper a phase-field approach for structural topology optimization for a 3D-printing process which includes stress constraint and potentially multiple materials or multiscales is analyzed. First order necessary optimality conditions are rigorously derived and a numerical algorithm which implements the method is presented. A sensitivity study with respect to some parameters is conducted for a two-dimensional cantilever beam problem. Finally, a possible workflow to obtain a 3D-printed object from the numerical solutions is described and the final structure is printed using a fused deposition modeling (FDM) 3D printer.

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