Modeling, Simulation and Optimization for Biomedical Applications

A. Caiazzo, I.E. Vignon-Clementel, Chapter 3: Mathematical Modeling of Blood Flow in the Cardiovascular System, in: Quantification of Biophysical Parameters in Medical Imaging, I. Sack, T. Schaeffter, eds., Springer International Publishing, Cham, 2018, pp. 45--70, (Chapter Published), DOI 10.1007/978-3-319-65924-4_3 .

P. Colli, G. Gilardi, J. Sprekels, Asymptotic analysis of a tumor growth model with fractional operators, Asymptotic Analysis, 120 (2020), pp. 41--72, DOI 10.20347/WIAS.PREPRINT.2625 .
Abstract
In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn--Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases.

C.K. Macnamara, A. Caiazzo, I. Ramis-Conde, M.A.J. Chaplain, Computational modelling and simulation of cancer growth and migration within a 3D heterogeneous tissue: The effects of fibre and vascular structure, Journal of Computational Science, 40 (2020), 101067, DOI 10.1016/j.jocs.2019.101067 .
Abstract
The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Since cancer cells can arise from any type of cell in the body, cancers can grow in or around any tissue or organ making the disease highly complex. Our research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modeling. We present a 3D individual-based model which allows one to simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent (a single cell, for example) is fully realised within the model and interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, for example, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells. The current state of the art of the model allows us to simulate tumour growth around an arbitrary blood-vessel network or along the striations of fibrous tissue.

P. Colli, G. Gilardi, J. Sprekels, A distributed control problem for a fractional tumor growth model, Mathematics - Open Access Journal, 7 (2019), pp. 792/1--792/32, DOI 10.3390/math7090792 .
Abstract
In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three selfadjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn--Hilliard type phase field system modeling tumor growth that goes back to Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24.) The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in the recent work Adv. Math. Sci. Appl. 28 (2019), 343--375 by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.

P. Colli, G. Gilardi, J. Sprekels, Well-posedness and regularity for a fractional tumor growth model, Advances in Mathematical Sciences and Applications, 28 (2019), pp. 343--375.
Abstract
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P. Colli, A. Signori, J. Sprekels, Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, pp. published online on 21.10.2019, urlhttps://doi.org/10.1007/s00245-019-09618-6, DOI 10.1007/s00245-019-09618-6 .
Abstract
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 setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fréchet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

G. Dong, M. Hintermüller, K. Papafitsoros, Quantitative magnetic resonance imaging: From fingerprinting to integrated physics-based models, SIAM Journal on Imaging Sciences, 2 (2019), pp. 927--971, DOI 10.1137/18M1222211 .
Abstract
Quantitative magnetic resonance imaging (qMRI) is concerned with estimating (in physical units) values of magnetic and tissue parameters, e.g., relaxation times $T_1$, $T_2$, or proton density $rho$. Recently, in [Ma et al., Nature, 495 (2013), pp. 187--193], magnetic resonance fingerprinting (MRF) was introduced as a technique being capable of simultaneously recovering such quantitative parameters by using a two-step procedure: (i) given a probe, a series of magnetization maps are computed and then (ii) matched to (quantitative) parameters with the help of a precomputed dictionary which is related to the Bloch manifold. In this paper, we first put MRF and its variants into perspective with optimization and inverse problems to gain mathematical insights concerning identifiability of parameters under noise and interpretation in terms of optimizers. Motivated by the fact that the Bloch manifold is nonconvex and that the accuracy of the MRF-type algorithms is limited by the ?discretization size? of the dictionary, a novel physics-based method for qMRI is proposed. In contrast to the conventional two-step method, our model is dictionary-free and is rather governed by a single nonlinear equation, which is studied analytically. This nonlinear equation is efficiently solved via robustified Newton-type methods. The effectiveness of the new method for noisy and undersampled data is shown both analytically and via extensive numerical examples, for which improvement over MRF and its variants is also documented.

S.P. Frigeri, C.G. Gal, M. Grasselli, J. Sprekels, Strong solutions to nonlocal 2D Cahn--Hilliard--Navier--Stokes systems with nonconstant viscosity, degenerate mobility and singular potential, Nonlinearity, 32 (2019), pp. 678--727, DOI 10.1088/1361-6544/aaedd0 .
Abstract
We consider a nonlinear system which consists of the incompressible Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We suppose that the viscosity depends smoothly on the order parameter as well as the mobility. Moreover, we assume that the mobility is degenerate at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with no-slip boundary condition for the (average) velocity and homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak-strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn-Hilliard equation, with a given velocity field, in the three dimensional case as well.

L. Heltai, A. Caiazzo, Multiscale modeling of vascularized tissues via non-matching immersed methods, International Journal of Numerical Methods in Biomedical Engineering, 35 (2019), pp. 3264/1--3264/32, DOI 10.1002/cnm.3264 .
Abstract
We consider a multiscale approach based on immersed methods for the efficient computational modeling of tissues composed of an elastic matrix (in two or three-dimensions) and a thin vascular structure (treated as a co-dimension two manifold) at a given pressure. We derive different variational formulations of the coupled problem, in which the effect of the vasculature can be surrogated in the elasticity equations via singular or hyper-singular forcing terms. These terms only depends on information defined on co-dimension two manifolds (such as vessel center line, cross sectional area, and mean pressure over cross section), thus drastically reducing the complexity of the computational model. We perform several numerical tests, ranging from simple cases with known exact solutions to the modeling of materials with random distributions of vessels. In the latter case, we use our immersed method to perform an in silico characterization of the mechanical properties of the effective biphasic material tissue via statistical simulations.

L.O. Müller, A. Caiazzo, P.J. Blanco, Reduced-order unscented Kalman filter with observations in the frequency domain: Application to computational hemodynamics, IEEE Transactions on Biomedical Engineering, 66 (2019), pp. 1269--1276, DOI 10.1109/TBME.2018.2872323 .
Abstract
Objective: The aim of this work is to assess the potential of the reduced order unscented Kalman filter (ROUKF) in the context of computational hemodynamics, in order to estimate cardiovascular model parameters when employing real patient-specific data. Methods: The approach combines an efficient blood flow solver for one-dimensional networks (for the forward problem) with the parameter estimation problem cast in the frequency space. Namely, the ROUKF is used to correct model parameter after each cardiac cycle, depending on the discrepancies of model outputs with respect to available observations properly mapped into the frequency space. Results: First we validate the filter in frequency domain applying it in the context of a set of experimental measurements for an in vitro model. Second, we perform different numerical experiments aiming at parameter estimation using patient-specific data. Conclusion: Our results demonstrate that the filter in frequency domain allows a faster and more robust parameter estimation, when compared to its time domain counterpart. Moreover, the proposed approach allows to estimate parameters that are not directly related to the network but are crucial for targeting inter-individual parameter variability (e.g., parameters that characterize the cardiac output). Significance: The ROUKF in frequency domain provides a robust and flexible tool for estimating parameters related to cardiovascular mathematical models using in vivo data.

J. Sprekels, H. Wu, Optimal distributed control of a Cahn--Hilliard--Darcy system with mass sources, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, pp. published online on 24.01.2019, urlhttps://doi.org/10.1007/s00245-019-09555-4, DOI 10.1007/s00245-019-09555-4 .
Abstract
In this paper, we study an optimal control problem for a two-dimensional Cahn--Hilliard--Darcy system with mass sources that arises in the modeling of tumor growth. The aim is to monitor the tumor fraction in a finite time interval in such a way that both the tumor fraction, measured in terms of a tracking type cost functional, is kept under control and minimal harm is inflicted to the patient by administering the control, which could either be a drug or nutrition. We first prove that the optimal control problem admits a solution. Then we show that the control-to-state operator is Fréchet differentiable between suitable Banach spaces and derive the first-order necessary optimality conditions in terms of the adjoint variables and the usual variational inequality.

C. Bertoglio, A. Caiazzo, Y. Bazilevs, M. Braack, M. Esmaily-Moghadam, V. Gravemeier, A.L. Marsden, O. Pironneau, I.E. Vignon-Clementel, W.A. Wall, Benchmark problems for numerical treatment of backflow at open boundaries, International Journal of Numerical Methods in Biomedical Engineering, 34 (2018), pp. e2918/1--e2918/34, DOI 10.1002/cnm.2918 .
Abstract
In computational fluid dynamics, incoming velocity at open boundaries, or backflow, often yields to unphysical instabilities already for moderate Reynolds numbers. Several treatments to overcome these backflow instabilities have been proposed in the literature. However, these approaches have not yet been compared in detail in terms of accuracy in different physiological regimes, in particular due to the difficulty to generate stable reference solutions apart from analytical forms. In this work, we present a set of benchmark problems in order to compare different methods in different backflow regimes (with a full reversal flow and with propagating vortices after a stenosis). The examples are implemented in FreeFem++ and the source code is openly available, making them a solid basis for future method developments.

L. Blank, A. Caiazzo, F. Chouly, A. Lozinski, J. Mura, Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), pp. 2149--2185, DOI 10.1051/m2an/2018063 .

A. Caiazzo, F. Caforio, G. Montecinos, L.O. Müller, P.J. Blanco, E.F. Toro, Assessment of reduced order Kalman filter for parameter identification in one-dimensional blood flow models using experimental data, International Journal of Numerical Methods in Biomedical Engineering, 33 (2017), pp. e2843/1--e2843/26, DOI 10.1002/cnm.2843 .
Abstract
This work presents a detailed investigation of a parameter estimation approach based on the reduced order unscented Kalman filter (ROUKF) in the context of one-dimensional blood flow models. In particular, the main aims of this study are (i) to investigate the effect of using real measurements vs. synthetic data (i.e., numerical results of the same in silico model, perturbed with white noise) for the estimation and (ii) to identify potential difficulties and limitations of the approach in clinically realistic applications in order to assess the applicability of the filter to such setups. For these purposes, our numerical study is based on the in vitro model of the arterial network described by [Alastruey et al. 2011, J. Biomech. bf 44], for which experimental flow and pressure measurements are available at few selected locations. In order to mimic clinically relevant situations, we focus on the estimation of terminal resistances and arterial wall parameters related to vessel mechanics (Young's modulus and thickness) using few experimental observations (at most a single pressure or flow measurement per vessel). In all cases, we first perform a theoretical identifiability analysis based on the generalized sensitivity function, comparing then the results obtained with the ROUKF, using either synthetic or experimental data, to results obtained using reference parameters and to available measurements.

C. Bertoglio, A. Caiazzo, A Stokes-residual backflow stabilization method applied to physiological flows, Journal of Computational Physics, 313 (2016), pp. 260--278.
Abstract
In computational fluid dynamics incoming flow at open boundaries, or emphbackflow, often yields to unphysical instabilities for high Reynolds numbers. It is widely accepted that this is due to the incoming energy arising from the convection term, which cannot be empha priori controlled when the velocity field is unknown at the boundary. In order to improve the robustness of the numerical simulations, we propose a stabilized formulation based on a penalization of the residual of a weak Stokes problem on the open boundary, whose viscous part controls the incoming convective energy, while the inertial term contributes to the kinetic energy. We also present different strategies for the approximation of the boundary pressure gradient, which is needed for defining the stabilization term. The method has the advantage that it does not require neither artificial modifications or extensions of the computational domain. Moreover, it is consistent with the Womersley solution. We illustrate our approach on numerical examples  - both academic and real-life -  relevant to blood and respiratory flows. The results also show that the stabilization parameter can be reduced with the mesh size.

A. Caiazzo, R. Guibert, I.E. Vignon-Clementel, A reduced-order modeling for efficient design study of artificial valve in enlarged ventricular outflow tracts, Computer Methods in Biomechanics and Biomedical Engineering, 19 (2016), pp. 1314--1318.
Abstract
A computational approach is proposed for efficient design study of a reducer stent to be percutaneously implanted in enlarged right ventricular outflow tracts (RVOT). The need for such a device is driven by the absence of bovine or artificial valves which could be implanted in these RVOT to replace the absent or incompetent native valve, as is often the case over time after Tetralogy of Fallot repair. Hemodynamics are simulated in the stented RVOT via a reduce order model based on proper orthogonal decomposition (POD), while the artificial valve is modeled as a thin resistive surface. The reduced order model is obtained from the numerical solution on a reference device configuration, then varying the geometrical parameters (diameter) for design purposes. To validate the approach, forces exerted on the valve and on the reducer are monitored, varying with geometrical parameters, and compared with the results of full CFD simulations. Such an approach could also be useful for uncertainty quantification.

A. Caiazzo, R. Guibert, Y. Boudjemline, I.E. Vignon-Clementel, Efficient blood flow simulations for the design of stented valve reducer in enlarged ventricular outflow tracts, Cardiovascular Engineering and Technology, 6 (2015), pp. 485--500.
Abstract
Tetralogy of Fallot is a congenital heart disease characterized over time, after the initial repair, by the absence of a functioning pulmonary valve, which causes regurgitation, and by progressive enlargement of the right ventricle and pulmonary arteries. Due to this pathological anatomy, available transcatheter valves are usually too small to be deployed in the enlarged right ventricular outflow tracts (RVOT). To avoid surgical valve replacement, an alternative consists in implanting a reducer prior to or in combination with a transcatheter valve. We describe a computational model to study the effect of a stented valve RVOT reducer on the hemodynamics in enlarged ventricular outflow tracts. To this aim, blood flow in the right ventricular outflow tract is modeled via the incompressible Navier--Stokes equations coupled to a simplified valve model, numerically solved with a standard finite element method and with a reduced order model based on Proper Orthogonal Decomposition (POD). Numerical simulations are based on a patient geometry obtained from medical imaging and boundary conditions tuned according to measurements of inlet flow rates and pressures. Different geometrical models of the reducer are built, varying its length and/or diameter, and compared with the initial device-free state. Simulations thus investigate multiple device configurations and describe the effect of geometry on hemodynamics. Forces exerted on the valve and on the reducer are monitored, varying with geometrical parameters. Results support the thesis that the reducer does not introduce significant pressure gradients, as was found in animal experiments. Finally, we demonstrate how computational complexity can be reduced with POD.

A. Caiazzo, G. Montecinos, L.O. Müller, E.M. Haacke, E.F. Toro, Computational haemodynamics in stenotic internal jugular veins, Journal of Mathematical Biology, 70 (2015), pp. 745--772.
Abstract
Stenosis in internal jugular veins (IJVs) are frequently associated to pathological venous circulation and insufficient cerebral blood drainage. In this work, we set up a computational framework to assess the relevance of IJV stenoses through numerical simulation, combining medical imaging, patient-specific data and a mathematical model for venous occlusions. Coupling a three-dimensional (3D) description of blood flow in IJVs with a reduced one-dimesional model (1D) for major intracranial veins, we are able to model different anatomical configurations, an aspect of importance to understand the impact of IJV stenosis in intracranial venous haemodynamics. We investigate several stenotic configurations in a physiologic patient-specific regime, quantifying the effect of the stenosis in terms of venous pressure increase and wall shear stress patterns. Simulation results are in qualitative agreement with reported pressure anomalies in pathological cases. Moreover, they demonstrate the potential of the proposed multiscale framework for individual-based studies and computer-aided diagnosis.

A. Caiazzo, I. Ramis-Conde, Multiscale modeling of palisade formation in glioblastoma multiforme, Journal of Theoretical Biology, 383 (2015), pp. 145--156.
Abstract
Palisades are characteristic tissue aberrations that arise in glioblastomas. Observation of palisades is considered as a clinical indicator of the transition from a noninvasive to an invasive tumour. In this article we propose a computational model to study the influence of genotypic and phenotypic heterogeneity in palisade formation. For this we produced three dimensional realistic simulations, based on a multiscale hybrid model, coupling the evolution of tumour cells and the oxygen diffusion in tissue, that depict the shape of palisades during its formation. Our results can be summarized as the following: (1) we show that cell heterogeneity is a crucial factor in palisade formation and tumour growth; (2) we present results that can explain the observed fact that recursive tumours are more malignant than primary tumours; and (3) the presented simulations can provide to clinicians and biologists for a better understanding of palisades 3D structure as well as glioblastomas growth dynamics

C. Bertoglio, A. Caiazzo, A tangential regularization method for backflow stabilization in hemodynamics, Journal of Computational Physics, 261 (2014), pp. 162--171.
Abstract
In computational simulations of fluid flows, instabilities at the Neumann boundaries may appear during backflow regime. It is widely accepted that this is due to the incoming energy at the boundary, coming from the convection term, which cannot be controlled when the velocity field is unknown. We propose a stabilized formulation based on a local regularization of the fluid velocity along the tangential directions on the Neumann boundaries. The stabilization term is proportional to the amount of backflow, and does not require any further assumption on the velocity profile. The perfomance of the method is assessed on a two- and three-dimensional Womersley flows, as well as considering a hemodynamic physiological regime in a patient-specific aortic geometry.

A. Caiazzo, J. Mura, Multiscale modeling of weakly compressible elastic materials in harmonic regime and application to microscale structure estimation, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12 (2014), pp. 514--537.
Abstract
This article is devoted to the modeling of elastic materials composed by an incompressible elastic matrix and small compressible gaseous inclusions, under a time harmonic excitation. In a biomedical context, this model describes the dynamics of a biological tissue (e.g. lung or liver) when wave analysis methods (such as Magnetic Resonance Elastography) are used to estimate tissue properties. Due to the multiscale nature of the problem, direct numerical simulations are prohibitive. We extend the homogenized model introduced in [Baffico, Grandmont, Maday, Osses, SIAM J. Mult. Mod. Sim., 7(1), 2008] to a time harmonic regime to describe the solid-gas mixture from a macroscopic point of view in terms of an effective elasticity tensor. Furthermore, we derive and validate numerically analytical approximations for the effective elastic coefficients in terms of macroscopic parameters. This simplified description is used to to set up an efficient variational approach for the estimation of the tissue porosity, using the mechanical response to external harmonic excitations.

TH.I. Seidman, O. Klein, Periodic solutions of isotone hybrid systems, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 18 (2013), pp. 483--493.
Abstract
Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work (Krasnosel'skiĭ-Pokrovskiĭ 1974) of theirs by defining a setting of hybrid systems with isotone switching rules for a partially ordered set of modes and then obtaining a periodicity result in that context. An application is given to a partial differential equation modeling calcium release and diffusion in cardiac cells.

M. Grote, V. Palumberi, B. Wagner, A. Barbero, I. Martin, Dynamic formation of oriented patches in chondrocyte cell cultures, Journal of Mathematical Biology, 63 (2011), pp. 757--777.
Abstract
Growth factors have a significant impact not only on the growth dynamics but also on the phenotype of chondrocytes (Barbero et al. , J. Cell. Phys. 204, pp. 830-838, 2005). In particular, as chondrocyte populations approach confluence, the cells tend to align and form coherent patches. Starting from a mathematical model for fibroblast populations at equilibrium (Mogilner et al., Physica D 89, pp. 346-367, 1996), a dynamic continuum model with logistic growth is developed. Both linear stability analysis and numerical solutions of the time-dependent nonlinear integro-partial differential equation are used to identify the key parameters that lead to pattern formation in the model. The numerical results are compared quantitatively to experimental data by extracting statistical information on orientation, density and patch size through Gabor filters.

A. Barbero, V. Palumberi, B. Wagner, R. Sader, M. Grote, I. Martin, Experimental and mathematical study of the influence of growth factors and the kinetics of adult human articular chondrocytes, Journal of Cellular Physiology, 204 (2005), pp. 830--838.

P. Colli, A. Signori, J. Sprekels, Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis, Preprint no. 2770, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2770 .
Abstract, PDF (451 kByte)
This paper concerns a distributed optimal control problem for a tumor growth model of Cahn--Hilliard type including chemotaxis with possibly singular anpotentials, 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 singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

M. Kantner, Th. Koprucki, Beyond just "flattening the curve": Optimal control of epidemics with purely non-pharmaceutical interventions, Preprint no. 2748, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2748 .
Abstract, PDF (3116 kByte)
When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple "flattening of the curve". Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.

J. Sprekels, F. Tröltzsch, Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth, Preprint no. 2721, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2721 .
Abstract, PDF (326 kByte)
In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L¹-norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.

J.A. Brüggemann, Elliptic obstacle-type quasi-variational inequalities (QVIs) with volume constraints motivated by a contact problem in biomedicine, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Berlin, August 5 - 8, 2019.

J.A. Brüggemann, Solution methods for a class of obstacle-type quasi variational inequalities with volume constraints, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Quasi-Variational Inequalities and Generalized Nash Equi-librium Problems (Part II)'', August 5 - 8, 2019, Berlin, August 7, 2019.

A. Caiazzo, Data assimilation in one-dimensional hemodynamics, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2019), Minisymposium 36 ``Data-Driven Computational Fluid Dynamics (Part 2)'', September 30 - October 4, 2019, Eindhoven University of Technology, Netherlands, October 1, 2019.

A. Caiazzo, Multiscale hybrid modeling and simulation of cancer growth within a 3D heterogeneous tissue, Canada-Germany Workshop Mathematical Biology and Numerics, June 24 - 26, 2019, Universität Heidelberg, June 26, 2019.

K. Papafitsoros, Generating structure non-smooth priors for image reconstruction, Young Researchers in Imaging Seminars, March 20 - 27, 2019, Henri Poincaré Institute, Paris, France, March 27, 2019.

K. Papafitsoros, Generating structure non-smooth priors for image reconstruction, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, August 5 - 8, 2019, Berlin, August 6, 2019.

J.A. Brüggemann, Path-following methods for a class of elliptic obstacle-type quasi-variational problems with integral constraints, 23rd International Symposium on Mathematical Programming (ISMP2018), Session 370 ``Variational Analysis 4'', July 1 - 6, 2018, Bordeaux, France, July 2, 2018.

A. Caiazzo, Mathematical modeling and simulations of geothermal reservoirs, Virtual Physiological Human Conference (VPH2018), September 5 - 7, 2018, University of Zaragoza, Spain, September 6, 2018.

A. Caiazzo, Robust open boundary conditions and efficient data assimilation in multiscale hemodynamics, International Symposium ``Modeling, Simulation and Optimization of the Cardiovascular System'', October 22 - 24, 2018, Universität Magdeburg, October 22, 2018.

A. Caiazzo, Towards the personalization of (1D) blood-flow simulations, University of Amsterdam, Computational Science Lab, Netherlands, September 21, 2018.

M. Hintermüller, M. Holler, K. Papafitsoros, A function space framework for structural total variation regularization in inverse problems, MIA 2018 -- Mathematics and Image Analysis, Humboldt-Universität zu Berlin, January 15 - 17, 2018.

K. Papafitsoros, A function space framework for structural total variation regularization with applications in inverse problems, VI Latin American Workshop on Optimization and Control (LAWOC 18), September 3 - 7, 2018, Quito, Ecuador, September 4, 2018.

A. Caiazzo, Estimation of cardiovascular system parameters from real data, 2nd Leibniz MMS Days 2017, February 22 - 23, 2017, Technische Informationsbibliothek, Hannover, February 22, 2017.

A. Caiazzo, Homogenization methods for weakly compressible elastic materials forward and inverse problem, Workshop on Numerical Inverse and Stochastic Homogenization, February 13 - 17, 2017, Universität Bonn, Hausdorff Research Institute for Mathematics, February 17, 2017.

A. Caiazzo, A comparative study of backflow stabilization methods, 7th European Congress of Mathematics (7ECM), July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 19, 2016.

A. Caiazzo, Backflow stabilization methods for open boundaries, Christian-Albrechts-Universität zu Kiel, Angewandte Mathematik, Kiel, May 19, 2016.

A. Caiazzo, Multiscale modeling of weakly compressible elastic materials in harmonic regime, Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Numerische Simulation, Bonn, May 21, 2015.