Modelling Material Injection Into Porous Structures Under Non-isothermal Conditions

Abstract

In this work, the Theory of Porous Media (TPM) is employed to model percutaneous vertebroplasty, a medical procedure in which acrylic cement is injected into cancellous vertebral bone. Previously, isothermal macroscale models have been derived to describe this material injection and the mechanical interactions which arise. However, the temperature of the injected cement is typically below the human body temperature, necessitating the extension of these models to the non-isothermal case. Following the modelling principles of the TPM and considering local thermal non-equilibrium conditions, our model introduces three energy balances as well as additional constitutive relations. If restricted to local thermal equilibrium conditions, our model equations are in agreement with other examples of TPM-based models. We observe that our model elicits physically reasonable behaviour in numerical simulations which employ parameter values and initial and boundary conditions relevant for our application. Noting that we neglect capillary effects, we claim our model to be thermodynamically consistent despite the employment of simplifying assumptions during its derivation, such as the Coleman and Noll procedure.

Figures (13)

• Figure 1: Injection of cement into a vertebral body, simplified boundary value problem (BVP) as well as an overview of the three constituents which are considered. The pore size is greatly exaggerated for readability
• Figure 2: Dimensionless pore geometry. See text for description
• Figure 3: Plots of dimensionless interface area against cement saturation for $q^F=0.6$ and $n^F=0.85$
• Figure 4: Discretised simulation domain $\Omega$ with indicated boundaries $\Gamma_\mathrm{A}$, $\Gamma_\mathrm{B}$ and $\Gamma_\mathrm{C}$
• Figure 5: Profiles of the cement (---) and marrow ($\cdots\:\!$) fluid saturation. Common results with Brooks-Corey (top) and linear relative permeability factors (bottom)