A constitutive model for lung mechanics and injury applicable to static, dynamic, and shock loading

Abstract

A continuum model for lung parenchyma is constructed. The model describes the thermomechanical response over a range of loading rates-from static to dynamic to shock waves-and a range of stress states, including isotropic expansion, triaxial extension, simple shear, and plane wave compression. Nonlinear elasticity, viscoelasticity, and damage are included, with the latter associated with changes of biological function as well as mechanical stiffness. A Gram-Schmidt decomposition of the deformation gradient leads to strain attributes that enter the thermodynamic potentials as state variables. A free energy function is designed for loading at low to moderate rates and tensile pressures, whereby the tissue response, with surface tension, is preeminent. An internal energy function is designed for wave propagation analysis, including shock waves, whereby compressibility of the air inside the alveoli is addressed via a composite stiffness based on a closed-cell assumption. The model accurately represents the response to triaxial loading, pressure relaxation, and dynamic torsion with relatively few parameters. Longitudinal wave speeds are reasonable for ranges of internal airway pressure and transpulmonary pressure. Airway pressure strongly affects the response to plane wave compression. Criteria for local injury and damage progression depend on a normalized energy density and its gradient, where the latter is paramount for impact problems involving fast pressure rises. Results suggest that local damage associated with edema is induced at load intensities much lower than those that would cause significant stiffness changes due to rupture, major tearing, or local collapse.

A large deformation multiphase continuum mechanics model for shock loading of soft porous materials

“…Furthermore, the sides and top are assumed to be impermeable (i.e., “undrained”) such that the pore fluid (air) is confined solely within the column. While we have full control over the boundary conditions at the ends of the column, we have chosen the impermeable boundary condition for comparative purposes to the model developed by Clayton et al, 46 Clayton and Freed, 54 Clayton, 73 Clayton and Freed 79 in which they assume occluded pore air. One could imagine the experimental analog to this type of boundary condition wherein a section of soft porous material is placed in an impermeable sleeve and then uniaxially loaded.…”

A Large Deformation Multiphase Continuum Mechanics Model for Shock Loading of Lung Parenchyma: Investigating the Effects of Fluid Viscous Stress

“…Outcomes of the current research are intended to inform macroscopic, single-phase models of the lung for 3-D modeling of much larger domains. [6][7][8][9][10][11] Mixture theory was first established at finite strain by Truesdell and Toupin. 12 Subsequent works by Bowen [13][14][15] and others [16][17][18] applied this rich continuum theory to porous media.…”

Laplace Stretch: Eulerian and Lagrangian Formulations

“…Laplace stretch, as it has been used in the literature to date, e.g., [1,2,3,4,12,5,6,13,8,14,11,7], derives from a Gram-Schmidt (or QR) decomposition of the deformation gradient F, where matrix Q is orthogonal, and matrix R is upper triangular. Given a coordinate system with base vectors ( e 1 , e 2 , e 3 ), we denote such a decomposition as F = RU , where R = R ij e i ⊗ e j has orthogonal components, and U = U ij e i ⊗ e j has uppertriangular components.…”

“…The corresponding upper-triangular stretch tensor was proven very appealing for modeling anisotropic hyperelastic materials by Srinivasa [3]. Other recent applications of the Lagrangian decomposition address shape memory polymers [4], anisotropic composites [5], biological membranes [6], and soft biological tissues [7]. Advantages and drawbacks of using the upper-triangular decomposition in constitutive models are discussed in these and related works [8].…”