Calculation of oxygen pressures in tissue with anisotropic capillary orientation. II. Coupling of two-dimensional planes

Abstract

The mathematical descriptions of oxygen transport in terms of two-dimensional diffusion in planes perpendicular to the capillaries can be coupled for an ensemble of planes to yield a three-dimensional description. In tissue with a distinct orienta tion of the oxygen supplying structures, the capillaries, this leads to a description of oxygen pressure in a whole tissue volume. Muscle tissue is an example of such tissue. The method allows calculation of oxygen pressure ( p 0 2) at any location in the volume. Tissue volume oxygen status can be assessed by constructing a p 0 2 histogram, gathering p 0 2's at a large number of locations in the tissue.

Maintenance of Heterogeneity of Capillary Spacing is Essential for Adequate Oxygenation in the Soleus Muscle of the Growing Rat

“…Tissue P O 2 was calculated as described previously ( 7 , 18 , 19 ). Briefly, this entailed an extension of the Krogh model now accounting for all N capillaries, where P O 2 at any location in a plane perpendicular to the capillary orientation is calculated as where Q is oxygen consumption, is the permeability (Krogh's diffusion coefficient) of the tissue, f () takes into account the gradient due to passive oxygen diffusion ( r 2 in the Krogh model), and A i is the area supplied by the i th capillary of radius r c i and location i ( 18 ).…”

“…We determined the response in two muscle regions differing from each other with respect to fiber type and the angiogenic response to hypoxia ( 12 ). Finally, we evaluated with a mathematical model ( 7 , 18 , 19 ) how changes in capillarization of the growing muscle may affect muscle oxygenation.…”

Image-based modelling of skeletal muscle oxygenation

“…This is mainly due to the lack of relevant 3D datasets and/or efforts to reduce computing time. 2D models generally come along with the assumption that oxygen diffusion in the longitudinal direction (z-direction) of the muscle can be neglected, to simplify the diffusion process into a 2D problem [139,140,143]. This is a valid assumption for straight vessels along the muscle's main axis in z-direction with parallel cross sections in the xy-plane of the tissue, as the drop of capillary oxygen partial pressure is very low compared with the gradient perpendicular to the vessel.…”

Theoretical Models of Microvascular Oxygen Transport to Tissue

“…A simplified version of this model was later presented 29 . Hoofd 35 36 presented a steady‐state model appropriate for O 2 transport by capillary arrays or networks having a preferential direction, as occurs in skeletal muscle. An analytical solution was first presented for tissue PO 2 in planes perpendicular to the capillaries, and then a numerical method was described for connecting successive planes to obtain the 3D PO 2 distribution.…”