In my talk I consider a new class of degenerate "parabolic" type ODE-PDE couplings arising in the modelling of life sciences. The long-time dynamics of solutions is studied in terms of their attractors. Some open problems will also be discussed.
We present a result on local well posedness for a highly nonlocal nonlinear diffusion–adhesion system [RZ]. Macroscopic systems of this type were previously obtained through upscaling [ZR] and can account for the effect of microscopic receptor binding dynamics in cell–cell adhesion. The system couples an integro PDE featuring degenerate diffusion of porous medium type and nonlocal adhesion...
Bone material can be seen as porous and fiberous, exhibiting the quasi-brittle type mechanical response, with some degree of plasticity. Softening plasticity and fracture mechanics lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point...
Sweeping processes are a class of evolution problems with unilateral constraints which were originally introduced by J.J. Moreau, motivated by problems in elastoplasticity and nonsmooth mechanics. Later they have then found applications in several diverse disciplines: economic theory, electrical circuits, crowd motion modeling, biology.
In his work, Moreau considered moving convex...
Qualitative and quantitative variations in mechanobiological markers, such as extracellular matrix stiffness, cell adhesion, and cellular Young’s modulus, are strongly correlated with physiological state and frequently associated with pathological conditions. Atomic force microscopy (AFM) enables mechanical testing at the single-cell level, but interpreting indentation data is challenging due...
Viscoelastic materials appear in a wide range of fields: engineering, biology, geophysics, etc., such as synthetic polymers, biological tissues and concrete. In this talk we discuss a mathematical model of nonlinear fractional viscoelastic materials within the context of infinitesimal strain theory under quasi-static situation. Constitutive relations of such a generalized fractional...
In photoacoustic tomography, biological tissue is illuminated with a short laser pulse of near infrared light. The absorbed energy creates a local pressure increase that propagates through the tissue, governed by the acoustic wave equation and can be measured on the boundary. From this measured time-series the initial pressure in the tissue is reconstructed, providing valuable information on...
The mechanical models describe many functions of populations, organisms and cells including their motility, proliferation, and morphogenesis. The mathematical modeling is crucial for description of biological cells, soft tissues, and multifunctional materials in various applications stemming from mechanical and environmental engineering, biological and medical sciences. For example, we refer...
