Thomas K. Goldstick

Mathematical models of the spatial distribution of retinal oxygen tension and consumption, including changes upon illumination

To better understand oxygen utilization by the retina, a mathematical model of oxygen diffusion and consumption in the cat outer, avascular retina was developed by analyzing previously recorded profiles of oxygen tension (PO2) as a function of retinal depth. Simple diffusion modelling of the oxygen distribution through the outer retina is possible because the PO2 depends only on diffusion from the choroidal and retinal circulations and on consumption within the tissue. Several different models were evaluated in order to determine the best one from the standpoints of their ability to represent the data and to agree with physiological reality. For the steady state one-dimensional diffusion model adopted (the special three-layer diffusion model), oxygen consumption was constant through the middle layer and zero in the layers near the choroid and near the inner retina. On the average, the oxygen consuming layer, as found by nonlinear regression for each profile, extended from about 75% to 85% of the retinal depth from the vitreous. This is a narrow band through the mid-region of the photoreceptors. Oxygen consumption of the entire avascular retina, determined from fitting eight PO2 profiles measured in light-adapted retinas, averaged 2.7 ml O2(STP)/(100 g tissue · min), while the value determined from fitting thirty-two PO2 profiles measured in dark-adapted retinas averaged 4.4 ml O2(STP)/(100 g tissue · min). Consumption in the light was thus only 60% of that in the dark. This suggests that the outer retina is at greater risk of hypoxic injury in the dark than in the light, a fuinding of considerable clinical significance.

Estimation of retinal oxygen transients from measurements made in the vitreous humor

Measurements of the oxygen tension in the vitreous humor close to the retina were used to estimate the values at the retinal surface in the cat. The retina was modelled as an infinite plane in a semi-infinite medium and the analytical solution was obtained for the transient following a change in breathing gas from room air to 95% O2/5% CO2. The solution was, compared with experimental measurements. Our measurements indicate that at steady state, breathing room air, the oxygen tension at the surface of the retina is between 15 and 20 mmHg and the gradient from it averages −2·8 mmHg/mm. Transient oxygen tension measurements at several locations within 1·7 mm of the retinal surface agreed well with our mathematical model which could then be used to predict the transient at the retinal surface. In this way, our model can be used to estimate retinal transients from measurements in the preretinal vitreous humor and the distortion introduced by not measuring on the retinal surface itself can be eliminated. The model predicts that the distortion would be relatively small if the electrode were less than about 100 μm from the retina, provided the retinal transient had a time constant greater than a minute. In the steady state, the error would be less than 0·3 mmHg at this distance. These results are not only important for oxygen but apply, with minor adjustments, to all small molecules and ions which passively diffuse from the retina into the vitreous humor.

Diffusion of Oxygen in Plasma and Blood

D in fresh human plasma has been found to decrease almost linearly with total protein content over a wide range of concentration and to vary only +/-4% in normals and +/-13% in abnormals. The average values for D in normal human plasma, at 25 and 37 degrees C, are 1.62 and 2.18 X 10(-5) cm2/sec respectively. For normal human blood at 42% hematocrit, the values of D, at 25 and 37 degrees C, are 1.20 and 1.62 X 10(-5) cm2/sec respectively.

Mass transport to walls of stenosed arteries: Variation with Reynolds number and blood flow separation☆

Abstract Abnormalities in the exchange of substances between the arterial wall and the blood flowing within the lumen may occur when blood flow is disturbed near vessel constrictions, bends and branches. We have studied these effects using a computer simulation of mass transport to the wall, of a simple solute dissolved in the blood flowing through an axisymmetric constriction in a straight, cylindrical vessel with constant wall concentration. Over the complete nonturbulent range of Reynolds numbers, the results show three distinct mass transport patterns which are related to three distinctly different blood flow patterns. These occur at Reynolds numbers less than, equal to, or greater than that required for blood flow separation and vortex formation. With the last two flow patterns, there are regions of both enhanced and impaired mass flux and, surprisingly, these occur in close proximity. This analysis was applied to oxygen to determine if there could be an impairment in oxygen transport to the wall at the sites of blood flow disturbance. Our results may explain the apparent predilection of specific sites along the arterial tree for the initiation and progression of atherosclerosis.

Carbon monoxide-induced arterial wall hypoxia and atherosclerosis☆

The elevated carbon monoxide level found in tobacco smokers has been suggested as one etiologic factor linking it with atherosclerosis. Unquestionably carbon monoxide does induce some arterial wall hypoxia, which has been established as an atherogenic factor, but without knowing the extent and location of this hypoxia the importance of this mechanism could not previously be assessed. Carbon monoxide acts both by inducing hypoxemia and shifting the oxyhemoglobin equilibrium curve, with these effects acting on the oxygen transport system from both the luminal blood and the vasa vasorum. We have studied this system using a computer simulation of the human arterial wall and found significant, mid-medial hypoxia with blood carbon monoxide levels routinely found in smokers. Because these levels fluctuate, the hypoxia they induce would be expected to be uncompensated by increased vascularization and therefore potentially represent a much more significant factor in atherogenesis than chronic hypoxia alone.

Significance of luminal plasma layer resistance in arterial wall oxygen supply.

Previous analyses of the arterial wall oxygen supply system have assumed that a cell-free layer of plasma next to the endothelium is the major transport barrier in the lumen. Using a computer simulation, we have quantitatively tested this assumption. Our results show that oxygen diffusion gradients extend significantly into the flowing blood well beyond any plasma layer and that the major luminal transport resistance lies in the flowing blood and not in the plasma layer. The simulation was also employed to compute the effect of a reported 50% drop in plasma oxygen diffusivity. This rather large reduction did significantly lower oxygen levels within the arterial wall tissue. Whether such large reductions in diffusivity ever actually occur in human plasma is a subject of current controversy.

Isovolemic hemodilution increases retinal tissue oxygen tension

Abstract•Background: Therapeutic isovolemic hemodilution has been reported to improve blood flow and oxygen delivery. Few reliable measurements have been made, however, showing the effect of hemodilution on tissue oxygen tension.• Methods: We measured retinal oxygen tension during experimental isovolemic hemodilution in normal cats. Polarographic oxygen microelectrodes were placed in the vitreous humor within 100–200 μm of the retinal surface.•Results: Tissue oxygen tension increased initially during isovolemic hemodilution to a maximum approximately 50% above baseline at approximately two thirds of the original hematocrit level. Hemodilution beyond this point to lower hematocrits caused a steady decline in tissue oxygen tension. Cardiac output measured in one cat undergoing isovolemic hemodilution increased as hematocrit was lowered, but the cardiac erythrocyte flux actually decreased steadily.•Conclusion: The observed increase in tissue oxygen tension with hemodilution appears to be explained by a lesser reduction in capillary than in systemic hematocrit, coupled with an increased capillary blood flow. The increase in tissue oxygen tension we observed could in part explain the clinically beneficial effects of hemodilution.

Spatial Variation of the Local Tissue Oxygen Diffusion Coefficient Measured in situ in the Cat Retina and Cornea

A method for measuring the local oxygen diffusion coefficient (D) in an intact tissue, in situ, in a living cat is described. Values of D were calculated from nonlinear regression analysis of the polarographic (turn-on) transients using a semi-empirical model for the retina and a theoretical one for the cornea. Two types of microelectrodes were employed: in the retina, ones with extremely short recesses; and in the cornea, bare metal needles. The local D in the cat retina was practically homogeneous with a mean of 1.97 +/- 0.11.10(-5) cm2/s, at its body temperature of 37-38 degrees C, 70.6 +/- 3.3 percent of that in isotonic saline at 37 degrees C. In the cat corneal stroma, at its normal temperature in situ of 33 degrees C, D was also virtually homogeneous with a mean of 1.19 +/- 0.20.10(-5) cm2/s, 42.8 +/- 7.3 percent of that in isotonic saline at 37 degrees C.

Effect of pulsatility on oxygen transport to the human arterial wall

Oxygen transfer in fully-developed, pulsating, laminar flow in rigid and distensible tubes was simulated as part of a study of vessel-wall hypoxia and atherogenesis. The model used for the computations is based on dimensions and flows in the human thoracic aorta. The pulsatile velocity fields employed are based on those of Womersley with the modification that the radial convection is written relative to the moving wall. Womersley-type pulsatility, superposed on an axial Poiseuille flow, was found to negligibly affect oxygen transport to the wall. Caution should be exercised, however, in extending this conclusion to the complex pulsatility in actual living vessels.

Spatial variation of aortic wall oxygen diffusion coefficient from transient polarographic measurements

AbstractPolarographic current transients following a voltage step (turn-on transient) were measured with bare cathodes (25 μm diameter) and shallowly recessed oxygen microelectrodes (<5μm diameter). Except for the initial part of the current transient, the experimental measurements were in excellent agreement with simple models in the literature, which predict an inverse relationship with