Cees Bos

Mathematical model of erythrocytes as point-like sources

A new approach to investigate the effect of pericapillary gradients, caused by the particulate nature of blood, on oxygen partial pressure (pO2) in tissue is presented. The blood erythrocytes are modeled as point-like sources, which makes the system independent of the geometry of the erythrocytes. This model is semi-analytical and is developed to estimate the pO2 far from the erythrocytes. It does so through calculation of the extraction pressure, which accounts for the capillary oxygen drop as compared to homogeneous blood. It is particularly useful to estimate pO2 in regions where the oxygen concentration is low. Simulations have been performed for a cylindrical tissue geometry and parameters are chosen for rat heart muscle. In accordance with the literature, for a fixed total oxygen supply low hematocrit values result in a lower pO2 at the border of the tissue cylinder than high values do. Also a decrease in hematocrit results in higher values for the extraction pressure. Finally, it was found that the effect of the particulate nature of blood is most distinct at low hematocrit values.

Treatment of failed open reduction for congenital dislocation of the hip. A 10-year follow-up of 14 patients.

Fourteen patients with 15 congenital dislocated hips with unsuccessful open reduction were referred for a secondary open reduction. All primary open reductions were incomplete due to an inadequate exposure and insufficient release of muscles and ligaments. The secondary open reduction was technically difficult. The patients were followed for 6-17 years. Six patients had no functional symptoms. Only three hips showed no radiographic deformity. In ten hips, femoral head necrosis was noted and two patients were left with shortening of 4 and 7 cm.

MODELLING ERYTHROCYTES AS POINT-LIKE O2 SOURCES IN A KROGHIAN CYLINDER MODEL

In modelling O2 transport to tissue, the capillary is often considered as a uniform oxygen source. Nevertheless, almost all the oxygen is supplied from the hemoglobin packed in single erythrocytes. Few literature models handle this particulate nature of blood O2 supply, and even fewer consider the consequences in a tissue pO2 model. Then, these particulate models are often too complicated or too laborious to be applied on large portions of tissue or on many tissue cases, as is needed for correct judgement of tissue oxygenation, e.g., as pO2 histograms. For a recent overview, see Popel (1989). Here, the simplest way of representing O2, sources, the point-like source, is worked out for handling erythrocytic O2 supply, and the consequences for tissue O2 considered referring to the simple basic configuration of the Krogh/Kety cylinder model.

The effect of blood flow on oxygen extraction pressures calculated in a model of pointlike erythrocyte sources for rat heart

A mathematical description of pericapillary oxygen gradients that takes into account the particulate nature of blood is possible in terms of erythrocytes as pointlike sources. The formulation in terms of quasi-stationary sources [1] is extended to account for moving erythrocytes. The extended model is semianalytical and can be used to estimate the extraction pressure (EP), which quantifies the effect on partial pressure of oxygen (pO2) in the tissue far from the erythrocytes. Simulations have been done for rat heart muscle tissue around a capillary. For low hematocrit (Hct; 20%) and low blood velocity EP is highest, higher than the pO2 drop in a surrounding typical tissue cylinder. This means that the impediment to O2 release close to the capillary can be larger than that to transport further into the tissue. Increasing the hematocrit decreases EP, that is, it facilitates O2 release. Increasing the blood velocity decreases EP at low Hct values but has the opposite effect at high Hct values (> 35%). For zero velocity, results are the same as with the quasi-stationary model.

Local Plasma Convection Can Be Important for Oxygen Release in Tissue Capillaries

Aroesty and Gross investigated the effect of mixing in gaps between two successive RBCs on the local mass transfer at the capillary wall. They found that the effect was insignificant. In this presentation it is shown that their conclusion results from their choice of boundary conditions. When boundary concentrations are chosen which are more similar to those in diffusional models the flux at the capillary wall increases significantly when mixing increases. The calculations presented indicate that mixing of plasma may enhance oxygen transport, although it is impossible to assess the physiological importance. To be able to investigate that, a model has to be developed that includes oxygen transport in both the RBCs and tissue, the oxygen release in the RBCs, and the oxygen consumption in the tissue.

The Effect of Capillary Blood Flow on the Oxygen Release into Rat Heart Tissue

Modelling of oxygen transport into tissue can be considered in two steps. The first step comprises the release from the oxygen carriers, the erythrocytes, in the streaming blood up to the capillary wall. The second step models the diffusional transport in the - stagnant -tissue, including the capillary wall. Many literature models only involve the second step. The first model by Krogh (1919) used the simplified geometry of a tissue cylinder around a centrally located capillary. Since, several extensions have been made to this model establishing the basis of calculation of pO2 in muscle tissue (Hoofd, 1992).

Plasma Mixing is Likely to Affect Capillary Oxygen Transport in Hard Working Rat Heart

The delivery of oxygen to tissue occurs primarily in the capillaries. Therefore a substantial part of the research on oxygen transport into the tissue has been directed towards oxygen release and transport in and around capillaries. The influence of the major determinants on this transport, such as diffusion coefficient, blood flow, reaction kinetics, has been well investigated. One of the possible influences on the oxygen transport in the capillaries was, seemingly, eliminated by Aroesty and Gross (1970). They investigated the effect of plasma mixing on oxygen transport in capillaries and showed that the net effect of mixing was negligible. The enhancement near the red blood cell (RBC) downstream of a small plasma volume was cancelled out near the upstream RBC. However, in a previous model (Bos et al., in press) we showed that the results of their study were influenced considerably by their choice of boundary conditions. In fact, one could easily show a 50% increase of the oxygen flux through the plasma to the tissue by altering the boundary conditions.