InterPore2025

Motion by beating of porous biofilms with heterogeneous rheology: Simulation and clinical assessment of therapy for cystic fibrosis

22 May 2025, 15:00

15m

Oral Presentation (MS21) Non-linear effects in flow and transport through porous media MS21

Speaker

Prof. Philippe Poncet (Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, Pau, France)

Description

Keywords and scope: mass transport in living systems, coupling of non-linear models, heterogeneous rheology, therapy monitoring, biological porous media, Stokes flow in moving geometries.

In this presentation we are interested in operational applications and new numerical approaches for modeling the heterogeneous mucus biofilm of human lungs for the monitoring of cystic fibrosis (CF) therapies. At an operational level, we aim at predicting whether a therapy has a significant impact of the mucociliary clearance or not, that is to say predicting the ability of the respiratory mucus to be functional (i.e. to be able to move efficiency toward the esophagus). By opposition, a non-functional mucus will not move sufficiently to clear the lung wall from allergens, toxic agents, viruses, bacteria and their residual products (DNA filaments and altered mucoïd elements).

The system studied is a mucus made of Newtonian PeriCiliary Liquid (PCL) and highly concentrated mucins produced by the goblet cells, flowing through and above the epithelium ciliated cells (a porous medium described at its pore-scale, ie resolving the cilia individually), as shown on the joint figure. The cilia vibration generates a mixture between the mucins and the PCL, leading by reaction to a polymerized mucus with a heterogeneous rheology (space a time variable). Among the rheological features such as visco-elasticity, visco-plasticity, yield stress and shear-thinning, we focus on this last one which has been shown to be the dominant feature leading to non-functional mucus [3]. This leads to two-kinds of non-linearity whose effects compete or cooperate and provide functional or non-functional mucus propulsion.

The first non-linearity is the non-Newtonian pseudo-stationary Stokes equation driving the mucus motion, written $-div(2\mu (c,D)D(u))=f-\mathrm{\nabla }p$, where $f$ is the driving force induced by the epithelial cell, $\mu$ is mucus viscosity, $D=(\mathrm{\nabla }u+\mathrm{\nabla }{u}^T)/2$ is the shear-rate of the velocity $u$, $p$ is the pressure, and the incompressibility is satisfied by $div(u)=0$.
Here $c(x,t)$ denotes the mucin concentration, and this Stokes equation is non-linear by means of the mucus shear-thinning rheology modeled by the Carreau law
${\displaystyle \mu (c,D)={\mu }_{\mathrm{\infty }}+({\mu }_0(c)-{\mu }_{\mathrm{\infty }}){(1+2\beta (c{)}^2|D{|}^2)}^{\frac{q(c)-2}{2}}}$

The second non-linearity concerns the mucin concentration which satisfies a transport-diffusion model ${\displaystyle {\partial }_{t}c+div(uc)-div(\sigma \mathrm{\nabla }(c))=0}$, since $u$ can be written as a function of $c$. Beside, it can also been noticed that the diffusion becomes $-div(\sigma {\epsilon }^{1+\tau }\mathrm{\nabla }({\epsilon }^{-1}c))$ when upscaling the epithelium described as a porous medium (of porosity $\epsilon (x,t)$ and tortuosity index $\tau$).

We will show that numerical simulations using dedicated semi-Lagrangian methods [1,2] give good agreement with clinical picture of cystic fibrosis patients [4], whose sputum provides the rheological parameters. New clinical data also exhibit that simulations and sputum rheology allows to track pathology evolution for the patients under the recent triple-therapies.

This project MucoReaDy is funded by French National Agency of Research under the grant number ANR-20-CE45-0022.