Mathematical models for viscoelastic biological matter
The increased availability of experimental data on the mechanics of cells and tissues has stimulated the emergence of the field of mechanobiology, in which mathematics and mechanics are combined to enhance our understanding of biological materials and of the processes involved in physiological activities. The development of mathematical models for such complex systems is challenging, but the possible outcomes in terms of our capability to diagnose and treat pathological conditions provide a strong motivation for pursuing this research direction.
The present project focuses on continuum mechanical models and on a careful analysis of their mathematical properties, with particular emphasis on the description of activation processes. These are important for the control of any physiological activity, as they bring to life what would otherwise be a passive aggregate of heterogeneous substances. Among the many active elements of the human body, we restrict attention to skeletal muscles and neuronal cells, as they feature a rich phenomenology.
The planned research includes both applied and theoretical aspects. On the practical side, its outcomes can open new possibilities in the context of digital twins for personalized medicine. On the mathematical side, a new class of models can stimulate studies of broad interest in continuum mechanics and analysis.
The research project is articulated in three units
Università Cattolica del Sacro Cuore (local unit coordinator: Giulia Giantesio)
Politecnico di Milano (local unit coordinator: Davide Riccobelli)
Within this research projects, the following publications have been produced by the PoliMi research unit:
In (Riccobelli, 2024), we have explored the role of elastocapillarity in the surface growth of tumor spheroids. This work combines mathematical modelling and experimental validation to investigate how elasticity and surface tension interact during tumor development. The study highlights the impact of these interactions on the mechanical state of spheroids and provides insights into the stress patterns observed during their growth.
In (Riccobelli et al., 2024), we have shown that brittle fracture is preceded by an elastic instability in elastic solids. This study reveals how geometrical and physical nonlinearities drives the onset of elastic instability that ultimately leads to crack formation. Using a combination of theoretical models and phase-field methods, the research offers insights into the mechanisms underlying crack nucleation and the transition to fracture.
In (Magri & Riccobelli, 2024), we addressed the modelling of initially stressed incompressible solids. Our work clarifies the structural properties that the energy functional should satisfy from a physical perspective and provides an improved formulation for its numerical approximation.
In (Cerrone et al., 2024), we proposed a patient-specific mathematical framework for predicting glioblastoma growth, integrating reduced order modeling and neural networks. Using neuroimaging data, we developed a computational pipeline that leverages model reduction techniques and neural networks to predict tumor evolution and assist clinical decision-making.
References
2024
Elastocapillarity-driven surface growth in tumour spheroids
Growing experimental evidence highlights the relevant role of mechanics in the physiology of solid tumours, even in their early stages. While most of the mathematical models describe tumour growth as a volumetric increase of mass in the bulk, in vitro experiments on tumour spheroids have demonstrated that cell proliferation occurs in a thin layer at the boundary of the cellular aggregate. In this work, we investigate how elasticity and surface tension interact during the development of tumour spheroids. We model the tumour as a hyperelastic material undergoing boundary accretion, where the newly created cells are deformed by the action of surface tension. This growth leads to a frustrated reference configuration, resulting in the appearance of residual stress. Our theoretical framework is validated through experimental results of tumour spheroid cutting. Similar to fully developed tumours, spheroids tend to open when subject to radial cuts. Remarkably, even newly formed spheroids, which lack residual stress, exhibit this behaviour. Through both analytical solutions and numerical simulations, we show that this phenomenon is driven by elastocapillary interactions, where the residual stress developed in grown spheroids amplifies the tumour opening. Our model’s outcomes align with experimental observations and allow us to estimate the surface tension acting on tumour spheroids.
@article{riccobelli2024elastocapillarity,author={Riccobelli, Davide},journal={arXiv preprint arXiv:2410.03344},title={Elastocapillarity-driven surface growth in tumour spheroids},year={2024}}
We argue that nucleation of brittle cracks in initially flawless soft elastic solids is preceded by a continuum instability which cannot be captured without accounting for geometrically and physical nonlinearities of the constitutive response. To corroborate this somewhat counterintuitive claim, we present a theoretical and numerical study of the simplest model where a homogeneous elastic body subjected to tension is weakened by a free surface which then serves as a site of crack nucleation. We show that in this prototypical setting, brittle fracture starts as a symmetry breaking elastic instability activated by softening and involving large elastic rotations. The implied bifurcation of the homogeneous elastic equilibrium is highly unconventional due to its extraordinary sensitivity to geometry, reminiscent of the transition to turbulence. We trace the development of the instability beyond the limits of continuum elasticity by using quasi-continuum theory allowing one to capture the ultimate strain localization indicative of the formation of actual cracks.
@article{riccobelli2024fracture,author={Riccobelli, Davide and Ciarletta, Pasquale and Vitale, Guido and Maurini, Corrado and Truskinovsky, Lev},doi={10.1103/PhysRevLett.132.248202},issue={24},journal={Physical Review Letters},pages={248202},publisher={American Physical Society},title={Elastic Instability behind Brittle Fracture},volume={132},year={2024}}
Modelling of initially stressed solids: structure of the energy density in the incompressible limit
This study addresses the modelling of elastic bodies, particularly when the relaxed configuration is unknown or non-existent. We adopt the theory of initially stressed materials, incorporating the deformation gradient and stress state of the reference configuration (initial stress tensor) into the response function. We show that for the theory to be applicable, the response function of the relaxed material is invertible up to an element of the material symmetry group. Additionally, we establish that commonly imposed constitutive restrictions, namely the initial stress compatibility condition and initial stress reference independence, naturally arise when assuming an initial stress generated solely from elastic distortion. The paper delves into modelling aspects concerning incompressible materials, showcasing the expressibility of strain energy density as a function of the deviatoric part of the initial stress tensor and the isochoric part of the deformation gradient. This not only reduces the number of independent invariants in the energy functional, but also enhances numerical robustness in finite element simulations. The findings of this research hold significant implications for modelling materials with initial stress, extending potential applications to areas such as mechanobiology, soft robotics, and 4D printing.
@article{magri2024modelling,author={Magri, Marco and Riccobelli, Davide},doi={10.1137/24M1670226},journal={SIAM Journal on Applied Mathematics},number={6},pages={2342--2364},publisher={SIAM},title={Modelling of initially stressed solids: structure of the energy density in the incompressible limit},volume={84},year={2024}}
Patient-specific prediction of glioblastoma growth via reduced order modeling and neural networks
Donato
Cerrone, Davide
Riccobelli, Piermario
Vitullo, Francesco
Ballarin, Jacopo
Falco, Francesco
Acerbi, Andrea
Manzoni, Paolo
Zunino, and Pasquale
Ciarletta
Glioblastoma (GBL) is one of the deadliest brain cancers in adults. The GBL cells invade the physical structures within the brain extracellular environment with patient-specific features. In this work, we propose a proof-of-concept for mathematical framework of precision oncology enabling rapid parameter estimation from neuroimaging data in clinical settings.
The proposed diffuse interface model of GBL growth is informed by neuroimaging data, periodically collected in a clinical study from diagnosis to surgery and adjuvant treatment. We build a robust and efficient computational pipeline to aid clinical decision-making based on integrating model reduction techniques and neural networks. Patient specificity is captured through the segmentation of the magnetic resonance imaging into a computational replica of the patient brain, mimicking the brain microstructure by incorporating also the diffusion tensor imaging data.
The full order model (FOM) is first discretized using the finite element method and later approximated by a reduced order model (ROM) adopting proper orthogonal decomposition (POD). Trained by clinical data, we finally use neural networks to map the parameter space of GBL evolution over time and to predict the patient-specific model parameters from the observed clinical evolution of the tumor mass.
@article{cerrone2024patientspecific,author={Cerrone, Donato and Riccobelli, Davide and Vitullo, Piermario and Ballarin, Francesco and Falco, Jacopo and Acerbi, Francesco and Manzoni, Andrea and Zunino, Paolo and Ciarletta, Pasquale},journal={arXiv preprint arXiv:2412.05330},title={Patient-specific prediction of glioblastoma growth via reduced order modeling and neural networks},year={2024}}
