Conference Agenda

Motile cilia are slender, flexible hair-like organelles that are used by many cells to move and manipulate the fluids that surround them. Their effectiveness for this task is evidenced by their usage across eukaryotic life, from unicellular organisms that use cilia for propulsion, up to complex, multicellular organisms including humans, where cilia play a fundamental transport role in vital organs. Cilia are essential to the protection of our lungs, allowing for the effective clearance of the mucus that provides a mechanical barrier to pathogens and pollutants.

Modelling cilia motion and the flow fields that they generate has been an active topic of research in biofluiddynamics since the inception of the field. Early work focused on understanding individual cilium motion using drag-based models, such as resistive force theory, or describing the flows generated by cilia arrays by imposing velocities on ciliated boundaries.

In this talk, I will describe our recent work that takes advantage of the growth of computing power to bring together these two modelling paradigms by considering large-scale arrays of coupled cilia, connecting resulting flow fields with collective motion. Our approach, stemming from the Lagrangian Mechanics of Active Systems (Solovev, A. and Friedrich, B.M., 2021. Lagrangian mechanics of active systems. The European Physical Journal E, 44, pp.1-15), allows for our computations to link directly with experimentally measured cilia beats, while also reducing the number of dynamic degrees of freedom and providing the model flexibility needed to probe additional physical characteristics, such as cilium elasticity. In addition to describing this framework, I will present recent results illustrating the emergent metachronal waves given by the model and how they provide new insights into cilia driven flows. While these results are linked primarily to studies involving model organisms, I will describe some perspectives and extensions of the model to study mucociliary clearance in the lung.

Mathematical modeling of the respiratory system provides a structured framework for analyzing various processes involved in breathing. This approach may be particularly relevant to improve diagnosis in the context of lungs pathologies such as asthma, COPD, fibrosis or emphysema where spontaneous ventilation is impaired. In these situations, patients are admitted to intensive care units where artificial ventilation assists them with their breathing effort by injecting air through the trachea, leading to possible lung injuries. To describe the ventilation process and understand how the lungs react to these large pressure variations, we propose a dynamic continuum mechanics model describing the lung deformations as those of a hyperelastic material. We consider the non-linear constitutive law developped in [1] in the steady state setting and generalize it to an unsteady setting. Furthermore, we couple it to a 0D representation of the bronchial tree, viewed as a resistive diadic tree as in [2]. We choose to preserve the equivalent resistance of the tree while reducing the total number of generations, in an effort to lower computational cost. Several terminal subdomains are then defined, representing the region of influence of a path through the bronchial tree. This coupling between the air flow through the bronchial generations and the lung parenchyma deformations adds a pressure gradient in the elastic equations representing the action of the air flow on the elastic media. This additional term can be written as a non-local non-linear dissipative term which depends only on the parenchyma deformations. The resulting equations can be viewed as a poromechanical model describing the air-parenchyma interaction and satisfying energy estimates. To discretize in time the resulting system, we consider a Newmark scheme with a finite element approximation for the space discretization. Due to the non-local term, the associated finite element matrix is a dense matrix obtained by the product of two time-dependent matrices, greatly increasing computation and memory costs, especially as the system is solved by a Newton algorithm. To overcome this limitation, we propose different numerical strategies as well as a local approximation of the resistive term. Both non-local and local models are first compared on a simplified 2D case. In particular, we focus on their ability to dissipate energy and reproduce the variations of the lung volumes. Next, normal breathing scenarios are simulated for a 3D realistic lung geometry considering some boundary conditions modeling the impact of surrounding elements such as the rib cage or the diaphragm. The proposed coupled models shall be further used in pathological situations in particular to explore possible lung injuries for patients under artificial ventilation.

[1] Patte C, Genet M, Chapelle D. A quasi-static poromechanical model of the lungs. Biomech Model Mechanobiol. 2022

[2] Pozin N, Montesantos S, Katz I, Pichelin M, Vignon-Clementel I, Grandmont C. A tree-parenchyma coupled model for lung ventilation simulation. Int J Numer Meth Biomed Engng. 2017

Mechanics and, more specifically, stress fields possibly play a crucial role in the development of pulmonary fibrosis. This work aims to provide clinicians with diagnostic and prognostic tools based on mechanical simulation. Personalisation of these tools is critical for clinical pertinence, thus requiring numerical techniques for real-time estimation of patient-specific mechanical parameters. This study presents a hybrid approach [1] that integrates the Proper Generalised Decomposition (PGD) [2] with deep learning techniques to develop real-time diagnostic and prognostic tools for clinicians, particularly in assessing compliance fields in fibrotic lungs.
The proposed method mitigates the curse of dimensionality inherent in parametric systems by employing a tensor decomposition. The building brick [3] of the proposed framework is based on the concept of HiDeNN [4]. Indeed, each mode of the tensor decomposition is represented by a sparse neural network with constrained weights and biases to replicate standard Finite Element Method (FEM) shape functions. This constraint enhances model interpretability and facilitates transfer learning, significantly accelerating the training process. Moreover, the model's architecture is directly determined by the mesh and the interpolation order, eliminating arbitrary choices and allowing mesh adaptation during the training stage.
Similarly to PINNs, the physics of the problem is incorporated into the loss function during unsupervised training. The training process involves solving a minimisation problem, similar to classical model reduction. However, automatic differentiation within the neural network framework allows for greater flexibility in addressing non-linearities, particularly when linearisation is difficult. The framework, therefore, offers a flexible framework for surrogate modelling in non-linear mechanics, as demonstrated through 1D, 2D and 3D benchmark problems that validate its robustness against analytical and numerical reference solutions.
As mentioned, the surrogate model is meant to be a digital twin of the lungs. Its parametrisation must, therefore, account for the shape variability of lungs from one patient to another. To that aim, a shape registration pipeline from CT images has been developed, from which a library of shapes can be obtained and parametrised. Each patient shape parameters can then be fed into the surrogate model to get a patient-specific solution.
References
[1] Daby-Seesaram, A., Skardova, K., & Genet, M. Finite Element Neural Network Interpolation: Hybridisation with the Proper Generalised Decomposition for surrogate modelling. Submitted,
[2] Chinesta, F., Ladeveze, P., & Cueto, E. (2011). A Short Review on Model Order Reduction Based on Proper Generalized Decomposition. Archives of Computational Methods in Engineering, 18(4), 395–404. https://doi.org/10.1007/s11831-011-9064-7
[3] Skardova, K., Daby-Seesaram, A., & Genet, M. Finite Element Neural Network Interpolation: Interpretable Neural Networks for Solving PDEs. Submitted,
[4] Zhang, L., Cheng, L., Li, H., Gao, J., Yu, C., Domel, R., Yang, Y., Tang, S., & Liu, W. K. (2021). Hierarchical deep-learning neural networks: Finite elements and beyond. Computational Mechanics, 67(1), 207–230. https://doi.org/10.1007/s00466-020-01928-9

Mechanical ventilation is a life-saving therapy for conditions such as acute respiratory distress syndrome (ARDS), requiring careful balancing of breathing effort reduction, gas exchange improvement, and prevention of alveolar overdistension. Achieving optimal settings is challenging due to inter-subject variability, frequent comorbidities, and competing therapy objectives. Computational lung models integrating patient-specific anatomy and respiratory mechanics offer promising personalized strategies. However, despite the emergence of novel models, validation against clinically relevant experimental data remains scarce, hindering their path toward clinical application.

This work presents a digital-twin framework that couples a three-dimensional nonlinear finite-element model of lung parenchyma with a zero-dimensional airway-tree flow network. We validate the framework using porcine data from five ARDS-model subjects. The lung geometries were reconstructed from end-expiratory CT scans. Airway geometries were segmented and extended through a generative branching algorithm to capture sub-voxel airway generations. The lung parenchyma was segmented into high-fidelity meshes, with regional porosity fields derived from CT Hounsfield units to define spatially variable permeability distributions.

Two constitutive models, a Yeoh-style hyperelastic model and an exponential model, were implemented and calibrated under volume-controlled ventilation to ensure physiologically plausible tissue strains and pressures. The incorporation of pre-strain into our simulations allows for the introduction of a baseline pressure, akin to PEEP in volume-controlled ventilation. These models were then applied to simulate two clinically relevant strategies: the low-tidal-volume ARDSnet protocol and airway-pressure-release ventilation (APRV). Simulated global signals of airway pressure, flow, and volume were generated, closely resembling typical clinical ventilator outputs. Simultaneously, the digital-twin framework resolved the regional mechanical response of lung parenchyma, generating detailed distributions of deformation, strain, stress, and other continuum-mechanics-derived indicators.

Model evaluation was performed at both global and regional scales. On a global scale, simulated airway pressure, volume, and flow were compared against experimental recordings for both volume- and pressure-controlled ventilation. Regionally, end-inspiratory aeration distributions were compared to those derived from CT images. Deformation fields were compared to image-registration-derived deformation maps, enabling assessment of differences between the image-registration reference standard and predictions from the models under varying conditions, including ventilation modes and constitutive models. Tidal ventilation distribution metrics were further compared to regional ventilation measurements from imaging sequences.

This integrated evaluation highlights the strengths and limitations of the current framework. Globally, the coupled poromechanical-airway model successfully reproduced key features of respiratory mechanics across ventilation modes. Regionally, discrepancies in predicted aeration and deformation underscore areas for model improvement, particularly in regions with pronounced heterogeneity and boundary interfaces, where airway branching variability influences local ventilation.

These insights will guide future refinements, including incorporating viscoelastic and surface-tension effects, enhanced heterogeneity mapping, and improved chest-wall and diaphragm interactions.