Roy W. Schubert

Axial diffusion and wall permeability effects in perfused capillary-tissue structures

Attempts to experimentally examine oxygen supply and distribution in the isolated perfused heart and brain have renewed interest in mathematical models of artificially perfused capillary-tissue structures. The need to understand histograms of PO2 measurements from these isolated-perfused organ studies (modified Lagendorf preparations) has required that existing mathematical models and their boundary conditions be re-examined in the context of these experiments. A unifying system of equations and boundary conditions are examined here for the purpose of studying the effects of anisotropic diffusion, and capillary vessel wall permeability on both the capillary and tissue substrate supply. The mathematical models are explored for parameters of physiologic interest, and some comparisons are made with experimental determinations. The comparisons with data suggest an anisotropic transport of oxygen in the tissue that is unexplained by known physiologic mechanisms.

Michaelis-Menten kinetics model of oxygen consumption by rat brain slices following hypoxia.

In the present study, we have measured partial pressure of oxygen (pO2) profiles through rat brain slices before and after periods of hypoxia (5 and 10 min) to determine its effect on tissue oxygen demand. Tissue pO2 profiles were measured through rat cerebral cortex slices superfused with phosphate buffer using oxygen (O2)-sensitive microelectrodes at different times in controls [40% O2 balance nitrogen (N2)], and at different times before and after 5 or 10 min of hypoxia (100% N2). A one-dimensional, steady-state model of ordinary diffusion with a Michaelis-Menten model of O2 consumption where the maximal O2 consumption (Vmax) and the rate at half-maximal O2 consumption (Km) were allowed to vary was used to determine the kinetics of O2 consumption. Actual pO2 profiles through tissue were fitted to theoretical profiles by a least-squares method. Vmax varied among penetrations in a control slice and among slices. Vmax seemed to decrease after hypoxic insult, but the change was not statistically significant. The Km value measured before hypoxia was lower than the first Km value measured after the end of hypoxia, indicating that hypoxia induced a compensatory change in the metabolic state of the tissue. Km increased immediately after both 5- and 10-min hypoxic insults and returned to normal after recovery for each case. It seems that during and immediately after short periods of hypoxia, Km increases from near zero but returns to normal values within a few minutes.

Mathematical evidence for flow-induced changes in myocardial oxygen consumption

The objective of this investigation was to aid in the determination of the mechanism by which oxygen consumption changes in proportion to coronary perfusion pressure or coronary blood flow. A mathematical model of oxygen transport and consumption in the isolated-perfused heart was developed, based on data from an autoregulating, cell-free perfused, externally paced, isovolumic feline heart preparation. The model features the unique combination of Michaelis-Menten kinetics, and one-dimensional (axial) diffusion in radially well-mixed tissue. An adaptive finite-difference integration routine was used to solve the resulting third order stiff two-point boundary value problem. A simplex minimization was employed to determine the parameter values that minimized the squared difference between the model and the experimental data in terms of tissue PO2 distribution (histograms). Different cases of the model representing pressure-induced, flow-induced, and “magnified” flow effects were run. The flow-dependent oxygen consumption version of the model produced a histogram squared error 30% lower than the pressure-induced version and 5% lower than any other case. The model and a critical review of the literature indicate that a flow-related mechanism is responsible for this phenomenon. Evidence also demonstrates that the Michaelis-Menten kinetics constant is not constant for different oxygen tensions.

On the computation of substrate levels in perfused tissues

Abstract The recent development of experimental techniques to study the isolated perfused heart and brain have renewed interest in the mathematical modeling of capillary-tissue structures. A new analytical representation is developed for the Krogh cylinder model for blood-tissue structures, and a scheme is presented for determining an asymptotic series approximation for this solution. This solution is explored for model parameters of experimental interest, and the contribution and effect of axial diffusion is demonstrated.

Electroglottograph as an additional source of information in isolated word recognition

Traditionally, speech recognition systems use only the acoustic speech signal (speech). However, the source of the signal and the way speech is produced and whether this information can aid in speech recognition needs to be investigated. The objective of this study was to assess the contribution of using the electroglottograph (EGG) as an additional source of information along with speech in an isolated word recognition system. The vocabulary consisted of 64 words, ranging from mono-syllabic words to words with four syllables. Two fully connected artificial neural networks were designed. One network (speech network) used only speech as its source of information. The other network (speech+EGG network) used EGG along with the acoustic speech signal as its source of information. The speech network had a peak recognition rate of 94.37%. The speech+EGG network had a peak recognition rate of 99.37%. Hence, the information provided by the EGG improved the performance of the speech recognition system by 5%.

An analytical model for axial diffusion in the Krogh cylinder.

The tissue level distribution of oxygen (O2) in the hemoglobinless perfused heart has been measured under controlled conditions with Whalen PO2 microelectrodes (Schubert, Whalen, and Nair, 19 78). In an attempt to understand tissue level details of O2 transport we compared that PO2 distribution to those predicted by mathmatical models. Models incorporating only radial transport in the tissue compared poorly with the measured histograms. Blum’s (1960) model with radial and axial diffusion was found to be in error (Schubert, 1976), and a correct solution had not been published. A unique analytic model was derived by modifying the radial transport phenomena so that the axial diffusional transport could be included. This model displays trends seen in the experimental data. Unfortunately the best match was obtained by increasing the value of the axial diffusion coefficient, D, considerably beyond the value accepted in the literature. This led to questions about the appropriateness of the radial transport assumptions which could be answered only be seeking the correct solution to the problem originally posed by Blum.

The Equivalent Krogh Cylinder and Axial Oxygen Transport

The Krogh-Erlang model has served as a basis of understanding of oxygen supply to resting and working muscle. Considerable discrepancy was found between pO2 microsensor data and results from that model. A modification was made to the transport mechanisms implied by the Krogh-Erlang model by averaging the tissue radially, using a mass-transfer coefficient to maintain radial transport, and adding axial diffusion in the capillary and tissue. This radially-averaged, axially distributed (RAAD) modified Krogh model is used to evaluate the hypothesis that axial transport is important in Krogh-geometry capillary-tissue structures. Analytic solutions for the modified model were developed. RAAD model histograms bear a striking resemblance to experimental data, while results from the classic model do not. The former has an SSE (sum of squared error) of 10.2 with respect to experimental histograms, while the Krogh model has an SSE of 238.6. The effect of using a radial mass-transfer coefficient was evaluated by comparing the RAAD model with a fully distributed model. It had been shown that the modified Krogh model predicts tissue level data well when the length-to-tissue radius ratio is 50. It was expected that the predictions would be degraded for smaller ratios and then the Krogh model would suffice. By supplying a fixed volume of tissue at different radius/length ratios, it will be shown that the modified Krogh model is superior in all aspects to the Krogh model. The results are slightly different from those of the distributed model, but these differences are limited to the first 10% of the arteriolar region. It is concluded that the RAAD model is a better overall predictor of oxygen distribution and may be useful in furthering our understanding of oxygen transport to tissue in hemoglobinless perfusion situations. We suggest that this radially-averaged, axially-distributed model be used in place of the classic Krogh cylinder model for all biological situations.

Evaluation of Myoglobin Function in the Presence of Axial Diffusion

Facilitation of oxygen transport by myoglobin has been assessed by many researchers. Yet, the models used in these studies often assume that radial diffusion is the primary transport mechanism in tissue. Axial diffusion is typically neglected. In this study, oxygen transport by myoglobin facilitation is added to a proven cardiac tissue model which contains axial diffusion in the tissue and capillary regions, the Radially-Averaged, Axially-Distributed (RAAD) model. Previous research has shown that the axial diffusion in the capillary and tissue regions becomes coupled, causing a reduction in the pO2 at the capillary inlet. The objective is to determine if this coupling effect increases the facilitation of oxygen transport by myoglobin. The RAAD model consists of non-interacting cylinders of tissue (Krogh cylinders), with each perfused by a central capillary. Derivation of the equations describing the RAAD model yields a stiff, fourth-order, non-linear, ODE, BVP. The equation set is solved numerically. Parameters for myoglobin concentration and diffusion coefficient are chosen to maximize myoglobin facilitation. The effect of myoglobin is assessed by observing changes in the pO2 profiles for the model with and without myoglobin. Also, the RAAD model is compared to experimental pO2 data to determine if the inclusion of myoglobin improves the model prediction. The computer simulations show that myoglobin does facilitate diffusion, but only to a small extent. The changes in the capillary pO2 profiles for the model with and without myoglobin are not significant, pO2 reductions are 0.8% at the inlet and 2% at the outlet. The model prediction is not substantially improved with the addition of myoglobin. The sum of squared error is reduced by 0.1%, from 5.6834 without myoglobin, to 5.6779 with myoglobin. The steady state solution of the RAAD model with myoglobin suggests that, in the presence of axial diffusion, facilitation of oxygen diffusion to tissue is not myoglobins primary function. No conclusion can be made about the transient function of myoglobin.

Two cytochrome oxygen consumption model and mechanism for carotid body chemoreception.

We have measured sinus nerve discharge, tissue PO2 and oxygen consumption (VO2) in cat carotid bodies under different experimental conditions using our recessed oxygen microelectrode. Our results indicate that the change in chemoreceptor activity with oxygen disappearance following blood flow occlusion can be related to a two cytochrome model for oxygen consumption as previously proposed by Mills and Jöbsis (1972).

Michaelis-Menten-Like Kinetics in the Krogh Tissue Cylinder

In 1978, Schubert, Whalen, and Nair published the results of a series of experiments involving a Krebs-Henseleit perfused, paced, isovolumic working cat heart. The stated primary purpose of their work was to estimate the distribution of tissue PO2 in the autoregu-lating heart at two levels of perfusion pressure. They also measured flow, oxygen consumption, oxygen extraction, and left ventricular function (peak left ventricular pressure and its derivative). Their results indicated that the distribution of tissue PO2 is under a fine degree of control. The only interval in the tissue PO2 histogram to show a highly significant difference for a change in perfusion pressure was the 0–5 mmHg range. It was also discovered that the autoregulating group exhibited good regulation of flow, oxygen extraction, oxygen consumption, and left ventricular function. Futhermore, their results suggested that oxygen is being shunted around the capillary bed, possibly by both a diffusive shunt and a convective shunt.