Conference Agenda

This paper is devoted to the multiscale modelling of transport processes in perfused tissues, such as liver, which are considered as multiporous media. The topic is important in the context of the perfusion diagnostics (in vivo), but also in the context of tissue engineering and regenerative medicine. In both these situations, the transport due to the advection-diffusion (A-D) is described at the tissue heterogeneity level, which can be treated in a dual way: 1) due to the hierarchical homogenization "micro-meso-macro", capturing the fluid-structure interaction (and accordingly also the A-D transport), or 2) using the "double porosity" ansatz which is based on a kind of "Double-Darcy model" (homogenization of the Darcy flow with large contrasts in the permeability and, accordingly, in the diffusivity and the advection velocity involved in the A-D transport equations. We consider both and compare both theses approaches in two situations:

Model-P: The contrast fluid (CF) enhanced CT perfusion test -- the A-D transport of the CF under the flow due to the standard blood perfusion;
Model-A: The CF (or other species) transport influenced by the acoustic wave propagation (with zero background flow) due the acoustic streaming (AS), which appears as a secondary effect of the acoustic waves propagating in the porous medium.

For Model-A, we consider periodic structures comprising solid skeleton and fluid-filled pores in which the CF is transported. Using the perturbation analysis, the nonlinear acoustics model is decomposed into linear, but coupled subproblems: the Acoustic Wave Propagation (AWP) problem, and the Acoustic Streaming Flow (ASF) problem. The two subproblems are treated by the asymptotic homogenization which yields the two-scale models for both these subproblems. The advection velocity field, as computed form the ASF subproblem, is involved in the A-D equation describing the transport of a dissolved species represented by the concentration. In the solid (or dual porosity - extra-vascular space, tissue parenchyma), a weak diffusion is considered, which The homogenization with the scaling ansatz is applied also to obtain the two-scale A-D transport model. A special care is devoted to the interface between the fluid channels (primary porosity, in general) and the dual porosity. We consider interface permeability and diffusivity (allowing for a jump in the pressure and concentration, respectively) which can depend on the ASF-generated shear stress). This enables to account for the "sonoporation" effect - the vessel wall becomes more permeable as the effect of the acoustic waves.

All the local and macroscopic models are implemented in the SfePy finite element code. Numerical examples illustrate the sensitivity of the delivery and transport on the microstructure geometry and other features.

Rohan, E., Naili, S., Homogenization of the fluid-structure interaction in acoustics of porous
media perfused by viscous fluid, Z. Angew. Math. Phys., vol 71, 137, 2020.

Rohan, E., Moravcová, F. Acoustic streaming in porous media – homogenization based two-scale modelling, 2024 J. Phys.: Conf. Ser. 2647 232009.

Rohan, E., Turjanicová, J., Lukeš, V. Multiscale modelling and simulations of tissue perfusion using the Biot-Darcy-Brinkman model, Computers & Structures, Volume 251, 2021, 106404.

This work presents a mathematical framework for modeling the transport of biological fluids and nutrients within vascularized tumours, incorporating microscale effects into a macroscopic representation. Tumour growth and progression are highly dependent on the exchange of fluids and nutrients with surrounding vasculature. However, the abnormal and heterogeneous nature of tumour-induced angiogenesis creates complex spatial variations in fluid flow and nutrient distribution, influencing the effectiveness of treatments such as chemotherapy and immunotherapy. Understanding these transport mechanisms is therefore crucial for improving predictive tumour models and developing more effective therapeutic strategies.

To capture these intricate dynamics, we employ asymptotic homogenisation, a widely-used upscaling technique designed to analyse and model multiscale systems. By exploiting the separation of scales, asymptotic homogenisation enables a systematic transition from detailed microscopic descriptions to simplified macroscopic models, while preserving essential microscale information encapsulated in effective parameters at a reduced computational cost. The model describes the tumour microenvironment as a double porous medium, consisting of both the tumour tissue and the embedded vasculature, which interact through fluid and nutrient exchange. The interstitial fluid, representing both extracellular fluids and cells, is assumed to be incompressible, with tumour growth accounted for through a volumetric source term proportional to nutrient availability and a non-convective mass flux driven by nutrient gradients. The governing equations describe fluid motion influenced by both pressure and nutrient concentration gradients, which reduce to Darcy’s law in absence of microscale variations. Nutrient transport is modeled through a coupled advection-diffusion-reaction system, where permeability and diffusivity tensors are derived from cell-level problems to account for microvascular geometry. The exchange of both fluids and nutrients between the tumour and vasculature is described using the Kedem-Katchalsky formulation, incorporating microscale transport properties into the macroscopic equations. This framework enables a computationally efficient representation of tumour-vasculature interactions while preserving critical microscale influences.

The proposed model serves as a powerful tool for simulating tumour growth and treatment response in a patient-specific context, as it can be adapted to realistic tumour geometries reconstructed from medical imaging. By integrating microscale vascular properties into macroscale tumour models, it enhances predictive accuracy and supports the development of personalized treatment strategies. While the current model assumes a static tumour domain and simplified proliferation mechanisms, it establishes a foundation for future advancements incorporating mechanical deformations, dynamic vascular remodeling, and therapy-induced modifications in the tumour microenvironment. These extensions will further bridge the gap between mathematical modelling and clinical applications, improving the model’s relevance for oncological research and treatment planning.

Modelling material failure is an open problem in continuum mechanics, with many modelling techniques filling the panorama of damage mechanics. Our research has focused on creating a model that can account for how microscopic changes effect the macroscopic process of material failure. To that end, we have developed a novel multiscale model of the damage phase field method. This was achieved via new analytical methods introduced for the upscaling of the damage phase field. As a result, we have a rigorous multiscale model of material damage, which can be employed to study a myriad of physical phenomenon. Including processes such as plasticity, cyclic damaging or sudden material failure. This work of course has many applications in the realm of material science, but we are more interested in clinical applications.

Our results were achieved three-fold. Firstly, we used the damage phase field method to develop a model of material failure across an arbitrary linear elastic composite material. Such a model allows us to approximate a wide variety of materials: including alloys, soft tissues and other engineered structures. Next, we employed asymptotic homogenisation to upscale our model into a computationally feasible macroscopic model. This macroscopic model encodes all the microscopic information about a materials microstructure into a set of effective coefficients. These effective coefficients are dependent on the solutions to a series of local cell problems which relate the macro and micro scales of our problem. Finally, we explored the question of how to solve our damage problem numerically. We found that a novel approach was required to calculate a solution. The damage phase field method has solutions depending on an optimisation problem which is not solved trivially. By developing a new numerical algorithm, we were able to implement effective code that allowed us to do two dimensional simulations of a trouser test. A trouser test is when we clamp a material in a rectangular reference configuration at one end and pull it apart at the other end. The resulting final configuration looks like a pair of trousers, hence the name. By studying the results of these simulations, we have found that a materials microscopic properties and geometry have a strong influence on material failure.

Our long-term goal is to apply our modelling methods to biological phenomena of soft tissue tearing. Namely, diseases such as aortic dissections and ACL ruptures. By applying a multiscale model of these diseases, we could potentially understand what microscopic changes in the body are making people more susceptible to these diseases. The clinical applications of these models would include predicting the occurrence of soft tissue tearing and preventing it, or a better understanding of the long-term effects of treatment. This would allow clinicians to make more informed decisions, leading to better patient outcomes.

Cells respond to their local environment through mechanotransduction, converting mechanical signals into a biological response, facilitating changes in cell function (e.g. cell growth, proliferation or differentiation). The cell cytoskeleton, particularly actomyosin stress fibres (SFs), and focal adhesions (FAs), which bind the cytoskeleton to the extra-cellular matrix (ECM), are central to this process, activating intracellular signalling cascades in response to deformation.
We present a novel bio-chemo-mechanical one-dimensional model to describe the formation and maturation of these mechanosensing structures, coupled through a positive feedback loop, and the associated cell deformation. In particular, we employ reaction-diffusion-advection equations to describe: the polymerisation of actin and bundling and activation of the resultant fibres; the formation and maturation of adhesions between the cell and substrate; and the activation of certain signalling proteins in response to FA and SF formation. A set of constitutive relations then connect the concentration of these key proteins to the mechanical properties of the cell cytoplasm and the ECM. In particular, we treat the cell as a Kelvin-Voigt viscoelastic material, with additional active stresses due to myosin II motor contractility. By connecting the cytoskeletal mesh scale to the microscale using discrete-to-continuum upscaling, our approach advances on the homogenised account of the cell provided by many existing models by rationally connecting the nanoscale and microscale features of cell-substrate adhesion and the cell cytoskeleton.
We employ this model to understand how cells respond to external and intracellular cues in vitro. For example, we explain how: dependent upon the mechanical properties of the surrounding ECM, non-uniform patterns of cell striation develop, leading to FA and SF localisation at the cell periphery; myosin II inhibition leads to disruption of SFs and, in turn, FAs; and nanoscale ligand patterning exerts control over microscale adhesion and cytoskeleton development. Having demonstrated the ability of the model to replicate experimental observations, the model provides a platform for systematic investigation into how cell biochemistry and mechanics influence the growth and development of the cell and associated changes in cell function, and facilitates prediction of internal cell measurements that are difficult to ascertain experimentally (e.g. stress distribution).

Finally, extending this model to two dimensions facilitates the incorporation of other key mechanosensing structures, including the stiff cell nucleus, and plasma and cortical membranes. With these additions, we consider the axisymmetric analogue of our one-dimensional model and conduct a linear stability analysis to investigate the stability of this axisymmetric configuration to various normal modes of deformation. By identifying non-axisymmetric modes with positive growth rates our model also reveals a possible mechanism for self-driven surface patterning of cells in vitro.

Lymph nodes (LNs) are essential components of the immune system, where lymph fluid, containing immune cells and antigens, is processed. Their structure consists of a porous lymphoid compartment (LC) and a surrounding thin subcapsular sinus (SCS) that allows free fluid flow. In this talk, we present a mathematical model, derived using the asymptotic homogenization technique, to capture the multiscale nature of fluid flow within the lymph node. We employ numerical simulations to investigate flow patterns, pressure distributions, and shear stress in detail. These results provide valuable insights into the mechanical environment of the lymph node, advancing our understanding of its role in immune function and offering a foundation for exploring therapies for lymphatic disorders.