Non-linear phenomena in oxygen transport to tissue

A coupled finite element model of tumor growth and vascularization

We present a model of solid tumor growth which can account for several stages of tumorigenesis, from the early avascular phase to the angiogenesis driven proliferation. The model combines several previously identified components in a consistent framework, including neoplastic tissue growth, blood and oxygen transport, and angiogenic sprouting. First experiments with the framework and comparisons with observations made on solid tumors in vivo illustrate the plausibility of the approach. Explanations of several experimental observations are naturally provided by the model. To the best of our knowledge this is the first report of a model coupling tumor growth and angiogenesis.

Modelling inert gas exchange in tissue and mixed-venous blood return to the lungs

Inert gas exchange in tissue has been almost exclusively modelled by using an ordinary differential equation. The mathematical model that is used to derive this ordinary differential equation assumes that the partial pressure of an inert gas (which is proportional to the content of that gas) is a function only of time. This mathematical model does not allow for spatial variations in inert gas partial pressure. This model is also dependent only on the ratio of blood flow to tissue volume, and so does not take account of the shape of the body compartment or of the density of the capillaries that supply blood to this tissue. The partial pressure of a given inert gas in mixed-venous blood flowing back to the lungs is calculated from this ordinary differential equation. In this study, we write down the partial differential equations that allow for spatial as well as temporal variations in inert gas partial pressure in tissue. We then solve these partial differential equations and compare them to the solution of the ordinary differential equations described above. It is found that the solution of the ordinary differential equation is very different from the solution of the partial differential equation, and so the ordinary differential equation should not be used if an accurate calculation of inert gas transport to tissue is required. Further, the solution of the PDE is dependent on the shape of the body compartment and on the density of the capillaries that supply blood to this tissue. As a result, techniques that are based on the ordinary differential equation to calculate the mixed-venous blood partial pressure may be in error.

The role of myoglobin in retarding oxygen depletion in skeletal muscle

Myoglobin retards the development of anoxia in a poorly perfused region of skeletal muscle by facilitating diffusion into this region from adjacent normally perfused regions and by releasing bound oxygen directly into the tissue. We examine these phenomena by analyzing a mathematical model of time-dependent myoglobin-facilitated oxygen transport. The governing equations are solved using similarity transformations and multiple-scale techniques. We find that when perfusion of a region is suddenly decreased, oxygen depletion is significantly retarded by direct release of myoglobin-bound oxygen into the tissue and that myoglobin-facilitated diffusion of oxygen from adjacent regions becomes significant at very low oxygen concentration.