Analysis of a spherical free boundary problem modelling granular biofilms

TL;DR

This paper models the formation and growth of multispecies granular biofilms using a spherical free boundary problem with PDEs, analyzing existence and uniqueness of solutions in the attachment regime.

Contribution

It introduces a continuum mechanics-based spherical free boundary model for multispecies biofilms, including novel PDE formulations and solution analysis.

Findings

Existence and uniqueness of solutions proved for the attachment regime.

The model captures advective transport, growth, and invasion phenomena.

Conversion of PDE system into an equivalent integral system using characteristics.

Abstract

A free boundary value problem related to the genesis of multispecies granular biofilms is presented. The granular biofilm is modelled as a spherical free boundary domain with radial symmetry. The proposed model is conceived in the framework of continuum mechanics and consists of: nonlinear hyperbolic PDEs which model the advective transport and growth of attached species that constitute the granular biofilm matrix; semilinear elliptic PDEs which govern the diffusive transport and conversion of nutrients; and semilinear elliptic PDEs describing the invasion phenomena and conversion of planktonic cells suspended in the surrounding environment. The evolution of the free boundary is governed by an ODE accounting for microbial growth, attachment, and detachment. By using the method of characteristics, the system of equations constituting the granular biofilm model is converted into an equivalent integral system. Existence and uniqueness of solutions are discussed and proved for the attachment regime using the fixed point theorem.