Calculation of oxygen pressures in tissue with anisotropic capillary orientation. I. Two-dimensional analytical solution for arbitrary capillary characteristics

Abstract

In tissue with a distinct orientation of the oxygen supplying structures, the capillaries, a mathematical description of oxygen transport is feasible in terms of two-dimensional diffusion in a plane perpendicular to the capillaries. Musclc tissue is an example of a highly anisotropic tissue. With some additional simplifying assumptions, a solution can be constructed in terms of capillary sources for arbitrary capillary characteristics, in particular, capillary locations. The solution includes facilitated diffusion by myoglobin in the tissue. For homogeneous tissue, the solution becomes explicit allowing direct calculation of tissue oxygen pressure at any location in a field of simple geometry (circular, rectangular). Also, the size of the area into which each capillary distributes its oxygen, the oxygen supply area, is readily calculated.

Capillary supply regions

“…(10). Hoofd [11] used a different boundary condition which is not unreasonable and effectively replaced the last term of Eq. (10) by a constant.…”

“…However, capillary supply regions were not considered. Hoofd et al [10] and Hoofd [11] required only the net flux on the boundary be zero. As we shall discuss later, such a solution may be non-unique.…”

Image-based modelling of skeletal muscle oxygenation

“…Figure 6 shows an example of the Voronoi tessellation (black lines) of a mouse soleus muscle. For tissue tessellated by such supply domains V i with areas A i , Hoofd [140] has generalized the Krogh cylinder solution for the oxygen diffusion to determine tissue oxygen partial pressure at location r to…”

“…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.…”

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

“…The complicated task of calculation of oxygen pressures in tissue is simplified considerably when a description in terms of two-dimensional diffusion is allowed [5]. Such a description seems appropriate for tissue with a distinct orientation, like muscle tissue, where the oxygen supply ing capillaries run largely in parallel [1].…”