A theoretical investigation of the increase in venous oxygen saturation levels in advanced glaucoma patients

Purpose: Vascular changes have been observed among glaucoma patients, but it is not yet known whether these vascular changes occur primary or secondary to glaucomatous damage. In this study, a theoretical mathematical model of the retinal vasculature is applied to a set of oximetry data obtained from healthy individuals and glaucoma patients and is used to propose possible explanations for the clinically observed increases in venous blood oxygen saturation in advanced glaucoma patients.Methods: Given clinical measurements of intraocular pressure (IOP), mean arterial pressure and arterial blood oxygen saturation from healthy persons and advanced (visual field mean defect (MD) ≥ 10 dB) primary open angle glaucoma (POAG, IOP > 21 mmHg) patients and advanced normal tension glaucoma (NTG, IOP ≤ 21 mmHg)patients, the model is used to predict the oxygen demand or Krogh cylinder tissue width that would yield the clinically-measured venous oxygen saturation in each population.Results: A decrease in retinal tissue oxygen demand (M0), an impairment in blood flow autoregulation, or a decrease in Krogh cylinder tissue width (d) can independently lead to increased venous saturation. The model predicts that a decrease in M0 or a decrease in d is more likely to yield the increased venous saturation levels observed in POAG patients, while impairing blood flow autoregulation with no change in M0 or d is more likely to yield the increased venous saturation levels observed in NTG patients.Conclusions: The combined theoretical and clinical model predictions suggest that the mechanisms leading to increased venous saturation might differ between POAG and NTG patients. The model predictions are used to hypothesize that a decrease in oxygen demand might be more relevant to the increase in venous saturation observed in advanced POAG, while impairment in autoregulation mechanisms might be more relevant to the increase in venous saturation observed in advanced NTG.

Measurement and Simulation of Water Transport During Freezing in Mammalian Liver Tissue

Optimization of cryosurgical procedures on deep tissues such as liver requires an increased understanding of the fundamental mechanisms of ice formation and water transport in tissues during freezing. In order to further investigate and quantify the amount of water transport that occurs during freezing in tissue, this study reports quantitative and dynamic experimental data and theoretical modeling of rat liver freezing under controlled conditions. The rat liver was frozen by one of four methods of cooling: Method 1—ultrarapid “slam cooling” (≥ 1000° C/min) for control samples; Method 2—equilibrium freezing achieved by equilibrating tissue at different subzero temperatures (−4, −6, −8, −10°C); Method 3°-two-step freezing, which involves cooling at 5°C/min. to −4, −6, −8, −10 or −20°C followed immediately by slam cooling; or Method 4—constant and controlled freezing at rates from 5–400°C/min. on a directional cooling stage. After freezing, the tissue was freeze substituted, embedded in resin, sectioned, stained, and imaged under a light microscope fitted with a digitizing system. Image analysis techniques were then used to determine the relative cellular to extracellular volumes of the tissue. The osmotically inactive cell volume was determined to be 0.35 by constructing a Boyle van’t Hoff plot using cellular volumes from Method 2. The dynamic volume of the rat liver cells during cooling was obtained using cellular volumes from Method 3 (two-step freezing at 5°C/min). A nonlinear regression fit of a Krogh cylinder model to the volumetric shrinkage data in Method 3 yielded the biophysical parameters of water transport in rat liver tissue of: Lpg = 3.1 X 10−13 m3/Ns (1.86 μ/min-atm) and ELP = 290 kJ/mole (69.3 kcal/mole), with chi-squared variance of 0.00124. These parameters were then incorporated into the Krogh cylinder model and used to simulate water transport in rat liver tissue during constant cooling at rates between 5–100°C/min. Reasonable agreement between these simulations and the constant cooling rate freezing experiments in Method 4 were obtained. The model predicts that the water transport ceases at a relatively high subzero temperature (−10°C), such that the amount of intracellular ice forming in the tissue cells rises from almost none (=extensive dehydration and vascular expansion) at ≤5°C/min to over 88 percent of the original cellular water at ≥50°C/min. The theoretical simulations based on these experimental methods may be of use in visualizing and predicting freezing response, and thus can assist in the planning and implementing of cryosurgical protocols.

Heat Transfer Normal to Paired Arterioles and Venules Embedded in Perfused Tissue During Hyperthermia

A numerical model of the heat transer normal to an arteriole-venule pair embedded in muscle tissue has been constructed. Anatomical data describing the blood vessel size, spacing, and density have been incorporated into the model. This model computes temperatures along the vessel walls as well as the temperature throughout the tissue which comprises an infinitely long Krogh cylinder around the vessel pair. Tissue temperatures were computed in the steady-state under resting conditions, while transient calculations were made under hyperthermic conditions. Results show that for both large- (1st generation) and medium-sized (5th generation) vessel pairs, the mean tissue temperature within the tissue cylinder is not equal to the mean of the arteriole and venule blood temperatures under both steady-state and transient conditions. The numerical data were reduced so that a comparison could be made with the predictions of a simple two-dimensional superposition of line sources and sinks presented by Baish et al. [1]. This comparison reveals that the superposition model accurately describes the heat transfer effects during hyperthermia, permitting subsequent incorporation of this theory into a realistic three-dimensional model of heat transfer in a whole limb during hyperthermia.

Model analyses of capillary growth and tissue oxygenation during hypoxia

Theoretical analyses were used to determine whether capillary growth is an adaptive response to hypoxia. Parameter values were obtained from models of transverse sections of muscles in which individual fibers were distributed in square-ordered arrays and capillaries were added to the perimeters of individual fibers in the arrays. Increasing the number of capillaries up to 2.0 per fiber increased hypoxic tolerance by 157% above that expected for a Krogh cylinder. However, increasing the number of capillaries from 2.0 to 4.0 per fiber increased hypoxic tolerance by only 18% and, assuming the entire perimeter of each fiber was perfused with blood, increased hypoxic tolerance by only 11% over the value obtained when capillary-to-fiber ratio was 4.0. Capillary growth during normal maturation may result in capillary-to-fiber ratios around 2.0, near the upper limit for producing marked changes in hypoxic tolerance. Therefore, capillary growth may not be an adaptive response to ambient hypoxia because there is little or no gas transport benefit derived from the additional capillaries.

A mathematical model for the freezing process in biological tissue

A mathematical model has been developed to study the process of freezing in biological organs. The model consists of a repetitive unit structure comprising a cylinder of tissue with an axial blood vessel (Krogh cylinder) and it is analysed by the methods of irreversible thermo­dynamics. The mathematical simulation of the freezing process in liver tissue compares remarkably well with experimental data on the structure of tissue frozen under controlled thermal conditions and the response of liver cells to changes in cooling rate. The study also supports the proposal that the damage mechanism responsible for the lack of success in attempts to preserve tissue in a frozen state, under conditions in which cells in suspension survive freezing, is direct mechanical damage caused by the formation of ice in the vascular system.

Microvessel surface area, density and dimensions in brain and muscle of the cephalopod Sepia officinalis

The microvasculature of brain and muscle in the cuttlefish Sepia was studied with stereological techniques to provide information about the surface area for exchange at the blood-tissue interface which was necessary for a parallel study of the permeability of the blood-brain barrier in Sepia . Microvessel density, length, dimensions and volume fraction, and the radius of the ‘Krogh cylinder’ of tissue supplied by each microvessel were also estimated. Vertical lobe (VL) and optic lobe (OL) of brain, outer collar valve muscle (VM) and tentacle muscle (TM) were analysed in 1 μm sections of aldehyde-fixed, Epon-embedded material. ‘Microvessels’ (diameter less than 20 μm) had a surface area density S v (in the order VL, OL, VM, TM) of 134, 176, 67.9 and 13.8 cm 2 cm -3 respectively. The numbers of microvessels per unit area tissue, Q A , were 211, 395, 157 and 43 mm -2 respectively. The length density of microvessels J V = 2 x Q A . The microvessel density was significantly greater in synaptic neuropil (NP) than neuron cell body (CB) zones. Total vessel volume density V V was 3.49, 4.73, 1.88 and 0.28%, in good agreement with previous estimates using intravascular tracers. Mean microvessel diameter d̄ was in the range 4.1-6.5 μm (mode 3.9-4.9 μm). The radius of the Krogh cylinder, R, was 28, 20, 32 and 61 μm. Calculations with the Krogh-Erlang equation show that brain and valve muscle are unlikely to be hypoxic under physiological conditions, while tentacle muscle may be. The vascular parameters correlate well with the known biochemistry of cephalopod tissues. This study represents a detailed analysis of the microvasculature in a complex invertebrate and permits useful comparisons with vertebrate tissues. Values for microvascular S V , Q A , J V and d̄ in Sepia brain are similar to those of the rat, while Sepia muscle vascularity is less than in the rat.