# Permeability Estimation

### Whole Organ Perfusion / Contrast Transport / Cathlab Diagnostics / Permeability Estimation

**Micro-Macro Vascular Coupling**

Although the poroelastic models do not directly make use of the microvascular topology, there are several reasons why having the ability to derive continuum macroscopic parameters from microvascular anatomy is useful. Permeability tensors calculated from such data can be used to parameterise the integrated perfusion model as well as in inferring changes to the distal parameters of the reduced 1D models, under diseased conditions.

Methodologies for deriving such macroscopic parameters have not been standardised and most previous work have focused on geomechanics applications. Prior to this work there was only one published method on deriving permeability tensors from a vascular network, and none based on actual vascular geometry. Thus we have pursued several different approaches. One way is via asymptotic homogenisation, which allows a mathematical reduction of a well-defined microvascular network. Alternatively, volume averaging of the vessel network via heuristic approaches was applied on capturing multi-compartmental characteristics. In particular, the porous flow model was combined with nonlinear fitting against direct flow simulations in the microvascular network. These approaches are outlined below.

**Permeability Estimation via Homogenisation**

The transmural variations in the permeability tensor for the capillary network was determined by an application of asymptotic homogenisation to a rat transmural microvascular network (Smith et al, 2014). Synthetic networks with geometrical and topological properties matching the experimental data were first produced, to yield a periodic network cell. Averaging procedure was applied on these cells to then derive the permeability tensors. The results showed a highly anisotropic ratio between the major local capillary direction and the orthogonal directions, and a gradual increase in capillary diameter from subepicardium to subendocardium was shown to translate to a 130% transmural rise in the principal permeability component.

**Permeability Estimation via Porous Flow Optimisation**

When characterising the vascular network as multiple hierarchical compartments (e.g. arterial, capillary, venous), a direct application of the asymptotic homogenisation faces an additional challenge since the division of the network into the compartments (based on diameter or otherwise) will yield a disconnected set of vessel segments. Thus instead, five different continuum multi-compartment parameterisation methods based on heuristic averaging were compared against a direct flow solution (Hyde et al, 2014). These methods primarily differed in the level of anatomical detail and the pressure-coupling fields utilised. The results showed that the most reliable parameterisation was achieved by porosity-scaled isotropic permeability assumption, and the anatomically-derived intercompartmental pressure-coupling fields.