Motivation Coupling of micro- and macro-models for complex flow and transport processes in biological tissue Katherina Baber, Bernd Flemisch, Rainer Helmig, Klaus Mosthaf

Department of Hydromechanics and Modeling of Hydrosystems

Free flow



Stokes equation

Porous media



Darcy ‘s law

Identification of dominating forces

→

dimensional analysis

Non-dimensional Stokes equation and transport equation

Goal:

coupling free flow with flow in porous media → focus on interface structure and processes

Application:

transvascular exchange of therapeutic agents 1

Transport equations Coupling conditions

2

Mechanical equilibrium:

Normal forces

Non dimensional transport equation combined with Darcy’s law

Mechanical equilibrium:

Tangential forces B

EAVERS

-J

OSEPH

-S

AFFMANN

-condition

A selection of dimensionless numbers:

Continuity of mass

Transvascular flow and transport processes (after Junqueira et al., 2002) → exchange processes strongly depend on wall structure and size and charge of the transported substance

Continuity of mass fractions

Free-flow region (ff) - Vasculature:

Assumption: steady laminar flow of a Newtonian fluid

Porous-medium regions (pm) – Capillary wall and tissue space:

Assumption: rigid porous media, continuous mobile Newtonian fluid phase

First results

Structure of the capillary wall: variety of para- and trans-cellular pathways, structure strongly dependent on anatomic location and physiological/pathological conditions → crucial to resolve the structure of the micro-vascular wall and the occurring transport processes

Current state Description of interface layer (capillary wall) as porous medium

→ average continuum description of the whole system (REV-scale) → → coupled model for free flow and flow in porous media single-phase compositional flow (blood/plasma + therapeutic agent) Distribution of pressure , x- and y-velocity and mass fractions of a dissolved substance

Drawbacks:

fails to resolve structure and processes at the interface volume averaging procedure not applicable to a thin heterogeneous structure like the capillary wall questionable if Beavers-Joseph-Saffman condition may be applied to biological problem setting

Work in progress Decoupling of interface layer and macro-model

→ new exchange and coupling conditions → different time and length scales possible → capillary wall as dominating structure for exchange processes resolved in more detail → inertial forces can be neglected in all three domains due to laminar viscous flow → advective transport processes dominate → flow directions differ strongly and velocities in transvascular pathways are not known resolve capillary wall on the micro-scale

Coupling concept for the micro-/macro-model:

→ coupling with the help of

mortar elements

to allow the transition from pore-scale to REV-scale model 3

Outlook

decide for partitioned or monolithic approach for each layer of mortar elements include electro-chemical processes and characteristics of transcvascular pathways derive 2D-interface by means of homogenization

Literature:

[1] Baber, K. (2009). Modeling the transfer of therapeutic agents from the vascular space to the tissue compartment (a continuum approach). Diploma thesis, Universität Stuttgart, Nupus preprint No. 2009/6 [2] Mosthaf K.; Baber, K.; Flemisch, B.; Helmig, R.; Leijnse, T.; Rybak, I. and Wohlmuth, B.: A new coupling concept for two-phase compositional porous media and single-phase compositional free flow. Submitted to Journal of Fluidmechanics (2010) [3] Balhoff, M. T. ; Thomas, S.G.; Wheeler, M. F.: Mortar coupling of pore-scale models. Comput Geosci. 12, 15-27 (2008)