Special Session 24: Optimal control and parameter estimation in biological models

Constrained optimization problems with applications to personalized medicine

Abramo Agosti University of Pavia Italy Co-Author(s):

Abstract: In this talk I will present different types of PDE-constrained optimization problems for multiphase tumour growth models which are of some interest for clinical applications. In particular, I will present the inverse identification problem of the initial condition from a single-point measurement at a final time for a Cahn-Hilliard type model describing tumour evolution in the a-vascular stage, and the parameters optimization problem for a multiphase model describing tumour growth in the vascular stage. I will show both analytical and numerical results, and present patient-specific test cases where the optimization problem is based on longitudinal neuroimaging data. This is a joint wotk with: Elena Beretta, Cecilia Cavaterra, Matteo Fornoni, Sabino Luzzi, Elisabetta Rocca

Solvability and optimal control for an epidemic propagation model with heterogeneous diffusion

Pierluigi Colli University of Pavia Italy Co-Author(s):

Abstract: Compartmental models are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels. In this talk, we are concerned with an epidemic Susceptible - Exposed - Infected - Recovered mathematical model in which the dynamics develops in a spatially heterogeneous environment. The four unknown functions s, e, i, r, which represent the four types of population, have to solve a nonlinear system of reaction-diffusion partial differential equations, complemented with homogeneous Neumann boundary conditions and initial conditions. The well-posedness of the problem is discussed and a control problems is studied with some details: for this problem the controls are the diffusion coefficients, which are supposed to be piece-wise constant. In fact, the existence of an optimal control can be shown and significant necessary conditions for optimality are derived. The tallk reports on joint works with Gianni Gilardi, Gabriela Marinoschi and Elisabetta Rocca.

On a diffuse interface model for the electrically-driven self-assembly of copolymers

Andrea Di Primio Politecnico di Milano Italy Co-Author(s):    Helmut Abels, Harald Garcke

Abstract: Self-assembling is an ubiquitous phenomenon in several applications, ranging from biology to materials science. In this talk, we consider a diffuse interface model describing a ternary system, constituted by a diblock copolymer and a homopolymer acting as solvent, interacting with an electric field. The dynamics of the ternary system is fully coupled with that of the electric field, hence the whole system is modeled by two Cahn--Hilliard--Oono equations for the copolymer blocks, accounting for long-range interactions; a classical Cahn--Hiliard equation for the homopolymer and, finally, the Maxwell equation for the electric displacement field. A multiphase singular potential is employed in order to ensure physical consistency. First, we show existence of global weak solutions in two and three dimensions. Uniqueness of weak solutions is estabilished in the constant mobility case, and a conditional result is given in the general case. Instantaneous regularization and long-time behavior are also investigated, the latter in the case of affine-linear electric permittivity, showing in particular that solutions converge to a single stationary state.

Maximal regularity and optimal control for a non-local Cahn-Hilliard tumour growth model

Matteo Fornoni University of Pavia Italy Co-Author(s):

Abstract: We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in the presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. Non-local cell-to-cell adhesion effects are included through a convolution operator with appropriate spatial kernels. First, we establish novel regularity results for such a model, by applying maximal regularity theory in weighted $L^p$ spaces. Such a technique enables us to prove the local existence and uniqueness of a regular solution, including also chemotaxis effects. By leveraging time-regularisation properties and global boundedness estimates, we further extend the solution to a global one. These results provide the foundation for addressing an optimal control problem, aimed at identifying a suitable therapy, which can guide the tumour towards a predefined target. Specifically, we prove the existence of an optimal therapy and, by studying the Fr\`echet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions.

Global Well-posedness of a Navier-Stokes-Cahn-Hilliard System with Chemotaxis and Singular Potential

Jingning He Hangzhou Normal University Peoples Rep of China Co-Author(s):    Hao Wu

Abstract: In this talk, we discuss a diffuse interface model that describes the dynamics of incompressible two-phase flows with chemotaxis effect. The PDE system couples the Navier-Stokes equations for the fluid velocity, a convective Cahn-Hilliard equation for the phase field variable with an advection-diffusion-reaction equation for the nutrient density. In the analysis, we consider a singular (e.g., logarithmic type) potential in the Cahn-Hilliard equation and prove the existence of global weak solutions in both two and three dimensions. In the two dimensional case, we establish a continuous dependence result that implies the uniqueness of global weak solutions. Furthermore, we prove the existence and uniqueness of global strong solutions that are strictly separated from the pure states over time in 2D.

Coefficient identification in nonlinear reaction-diffusion systems

Barbara Kaltenbacher University of Klagenfurt Austria Co-Author(s):    William Rundell

Abstract: Reaction-diffusion equations / systems appear in many applications. Often the coefficients or the nonlinearity itself is unknown and for the sake of generality one aims for a non-parametric form of the nonlinearity. In this talk we consider inverse problems of recovering potentials and/or state-dependent source terms in a reaction-diffusion system from overposed data consisting of the values of the state variables either at a fixed finite time (census-type data) or a time trace of their values at a fixed point on the boundary of the spatial domain. The basic idea of an iteration scheme that can be applied in many cases relies on projecting the data onto the observation manifold. We can then express those parts of the differential operators in the PDE that are tangential to this manifold via the data; and those parts that are perpendicular to the manifold via PDE solutions. This leads to a fixed point formulation and thus to a reconstructive method and we shall demonstrate its effectiveness by several illustrative examples.

Optimal control for an epidemic model

Gabriela Marinoschi Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy Romania Co-Author(s):    Gabriela Marinoschi

Abstract: We present a general problem of optimally controlling of an epidemic outbreak of a disease structured by age since exposure, with the aid of two types of control instruments, namely social distancing and vaccination. We prove the existence of at least one optimal control pair, derive the first-order necessary conditions for optimality and prove some useful properties of such optimal solutions. This general model can be specialized to include a number of subcases relevant for epidemics (e.g., like COVID-19), such as, the arrival of vaccines in a second stage of the epidemic, and vaccine rationing, making social distancing the only optimizable instrument in the first stage. The control problem takes also into account the indirect epidemic cost, namely the broader societal and economic cost due to the impact of social distancing on overall social and relational activities. The presentation is based on a joint paper with Alberto D`Onofrio, Mimmo Iannelli and Piero Manfredi.

Nonlinear oscillations in Fluid Mechanics

RICCARDO MONTALTO University of Milan Italy Co-Author(s):

Abstract: In this talk I shall discuss some recent results about the construction of quasi-periodic waves in Euler equations and other hydro-dynamical models in dimension greater or equal than two. I shall discuss quasi-peridic solutions and vanishing viscosity limit for forced Euler and Navier-Stokes equations and the problem of constructing quasi-periodic traveling waves bifurcating from Couette flow (and connections with inviscid damping). Time permitting, I also discuss some results concerning the construction of large amplitude quasi-periodic waves in MHD system and rotating fluids. The techniques are of several kinds: Nash-Moser iterations, micro-local analysis, analysis of resonances in higher dimension, normal form constructions and spectral theory.

Lipid rafts formation on cell membranes: modeling and mathematical analysis

Andrea Poiatti University of Vienna Austria Co-Author(s):    Helmut Abels, Harald Garcke

Abstract: In this talk I would like to present a model concerning lipid rafts formation on cell membranes (lipid bilayers). This phenomenon consists in the separation of lipids composing the cell membrane into two immiscible liquid phases, leading to the formation of heterogeneous liquid-ordered phase platforms (rafts). These rafts are believed to play important roles in the biology of the cell. I will focus on a diffuse interface model for incompressible viscous two-phase fluids with different densities, known as Abels-Garcke-Gr\{u}n model, over an evolving surface. After briefly showing the derivation of the model, I will introduce a suitable framework of evolving families of Banach and Hilbert spaces, and explain some recent results concerning the well-posedness of strong solutions to the problem, when the surface evolution is a priori prescribed. Namely I will first focus on the existence of a local strong solution, which is separated from pure phases, and then on how to extend this solution to a global-in-time unique separated strong solution. I will also present some new techniques for obtaining the validity of the strict separation property from pure phases on two-dimensional surfaces, under very weak assumptions on the behavior of the singular potential.

Analysis and simulations of a stochastic phase-field model for tumour growth

Luca Scarpa Politecnico di Milano Italy Co-Author(s):    Marvin Fritz

Abstract: We consider a stochastic phase-field model for tumour growth, coupling a Cahn-Hilliard equation for the order parameter, i.e. the difference in volume fractions between the tumour and healthy cells, with a reaction-diffusion equation for the nutrient. Both equations take into account the possible unpredictable model-oscillations via suitable stochastic forcing terms. First, the mathematical analysis of the system is performed in wide generality, including non-constant mobilities and chemotaxis. Secondly, numerical computations are performed in order to visualise the effect of the noise on the tumour growth and shape. In conclusion, possible developments towards optimal control problems are discussed. This study is based on a joint work with Marvin Fritz (Johann Radon Institute, Linz, Austria).

Optimal control of Cahn-Hilliard-Keller-Segel tumor growth models

Andrea Signori Politecnico di Milano Italy Co-Author(s):

Abstract: This presentation examines optimal control problems for two Cahn-Hilliard-Keller-Segel tumor growth models that capture key biological processes such as chemotaxis, angiogenesis, and nutrient consumption. These models combine a Cahn-Hilliard system for tumor and healthy cell segregation with a Keller-Segel equation for nutrient dynamics and chemotaxis. The goal is to minimize deviations in tumor cell distribution from target configurations while optimizing control costs. We establish the existence of optimal controls and derive first-order necessary optimality conditions under suitable assumptions.