Remodeling and growth of living tissue: a multiphase theory

Abstract

A continuum triphase model (i.e., a solid filled with fluid containing nutrients) based on the theory of porous media (TPM) is proposed for the phenomenological description of growth and remodeling phenomena in isotropic and transversely isotropic biological tissues. In this study, particular attention is paid on the description of the mass exchange during the stress-strain-and/or nutrient-driven phase transition of the nutrient phase to the solid phase. In order to define thermodynamically consistent constitutive relations, the entropy inequality of the mixture is evaluated in analogy to Coleman and Noll (Arch Ration Mech Anal 13: [167][168][169][170][171][172][173][174][175][176][177][178] 1963). Thereby, the choose of independent process variables is motivated by the fact that the resulting phenomenological description derives both a physical interpretability and a comprehensive description of the coupled processes. Based on the developed thermodynamical restrictions constitutive relations for stress, mass supply and permeability are proposed. The resulting system of equation is implemented into a mixed finite element scheme. Thus, we obtain a coupled calculation concept to determine the solid motion, inner pressure as well as the solid, fluid and nutrient volume fractions.

Perspectives on biomechanical growth and remodeling mechanisms in glaucoma

“…The Theory of Porous Media was used to account for growth based on both the state of mechanical stress and the availability of nutrients for mass exchange. The triphasic model was successfully used to investigate wound healing, stent restinosis occurring after angioplasty, bone remodeling, and topology optimization of organic material (Ricken et al, 2007; Ricken and Bluhm, 2009, 2010). It may prove useful to extend the triphasic concept to include the nutrient and enzyme concentrations in the tissue and to link the model with a microstructural-based growth formulation (Grytz et al, 2011b) to model connective tissue migration and loss in glaucoma.…”

Modeling sample/patient‐specific structural and diffusional responses of cartilage using DT‐MRI

“…) where R F is a positive definite material parameter tensor representing the intrinsic hydraulic resistance of the cartilage solid matrix and b is the body force per unit mass. Using , the fact that the motions of both solid and fluid are connected by the interaction forces , and considering thermodynamic restrictions (see, e.g., ), we propose the anisotropic intrinsic permeability of the cartilage solid matrix K F as (cf. ) where the permeability depends on the deformation and is characterized using the initial Darcy permeability k 0 S [m 4 /Ns] and m , a dimensionless material parameter controlling the general isotropic deformation dependence of the permeability (also see, e.g., ).…”

Transverse isotropic flow in biphasic materials

“…The motions of both the solid and the fluid are connected by the interaction forcesp F = −p S . From the evaluation of the entropy inequality, see RICKEN & BLUHM [2], for the constitutive modeling we gain the restriction forp F and postulate the following relation for the permeability tensor S F in case of transverse isotropic permeability:…”

“…Now, the balance equation of momentum (1) 2 will be evaluated for the fluid phase. Under consideration of (2) 1,2 , where the restrictions gained from the entropy inequality α F0 α F1 + α F3 ≥ 0 have to be insured, see RICKEN & BLUHM [2], as well as the assumptions x F = o (quasi static description) and T F E = 0 (non viscous fluid), we derive (1) 2 as…”