An implicit elastic theory for lung parenchyma

Abstract

The airways and parenchyma of lung experience large deformations during normal respiration. Spatially accurate predictions of airflow patterns and aerosol transport therefore require respiration to be modeled as a fluid-structure interaction problem. Such computational models in turn require constitutive models for the parencyhma that are both accurate and efficient. Herein, an implicit theory of elasticity is derived from thermodynamics to meet this need, leading to a generic template for strain-energy that is shown to be an exact analogue for the well-known Fung model that is the root of modern constitutive theory of tissues. To support this theory, we also propose a novel definition of Lagrangian strain rate. Unlike the classic definition of Lagrangian strain rate, this new definition is separable into volumetric and deviatoric terms, a separation that is both mathematically and physically justified. Within this framework, a novel material model capable of describing the elastic contribution of the nonlinear response of parenchyma is constructed and characterized against published data.

On elastic solids with limiting small strain: modelling and analysis

“…While most nonstandard implicit models involve both the stress and its time derivatives as well as the appropriate kinematical variable and its time derivative, there are materials for which one has an implicit relationship between just the stress and an appropriate kinematical quantity; the motivation for the need for such models stems from the existence of viscous fluids whose viscosity depends on the mean normal stress (such fluids are sometimes referred to as piezo-viscous fluids, see Szeri [69]). Similarly, implicit models arise naturally within the context of polymeric fluids with pressure dependent viscosity (see Singh and Nolle [66], McKinney and Belcher [36]), geological materials, which are viscoelastic fluids that have material moduli that depend on the mean normal stress (see the discussion and references in Karra et al [30]), certain biological elastic solids (see the discussion in Freed [22] and Freed and Einstein [23]), and elastomeric solids whose material moduli depend on the mean normal stress (see the discussion in Rajagopal and Saccomandi [56]). …”

Predicted airway obstruction distribution based on dynamical lung ventilation data: A coupled modeling‐machine learning methodology

“…Future work could include an inverse problem to determine both the plugs distribution and mechanical properties based on 4D data image registration. Some in-vivo techniques aim at determining the parenchyma mechanical properties based on image analysis [45], though in cases where the ventilation is strongly affected by pathological patterns within the tree, the tree-parenchyma coupling shall be taken into account. Indeed, some regions with normal mechanical behavior may not inflate because they are irrigated by constricted paths.…”

A Novel Approach to the Description of Constitutive Relations

“…Implicit constitutive relations are the natural class of response relations to describe a very large class of materials, polymeric fluids, which exhibit pressure 1 dependent material properties [see Singh and Nolle (1959) and McKinney and Belcher (1963)], geological materials, biological solid matter, such as DNA and collagenous material [see Freed and Einstein (2013);Freed (2014); Freed et al (2014); Freed and Rajagopal (2016)], elastomeric solids whose material moduli depend on the mean normal stress [see the discussion in Rajagopal and Saccomandi (2009)], magneto-elastic bodies [see Bustamante and Rajagopal (2015)], and electro-elastic bodies [see Rajagopal (2012, 2013)], to name some of them.…”