A mathematical model of tumor oxygen and glucose mass transport and metabolism with complex reaction kinetics

Hypoxia imparts radioresistance to tumors, and various approaches have been developed to enhance oxygenation, thereby improving radiosensitivity. This study explores the influence of kinetic and physical factors on substrate metabolism in a tumor model, based on a Krogh cylinder. In tissue, aerobic metabolism is assumed to depend on glucose and oxygen, represented by the product of Michaelis-Menten expressions. For the base case, an inlet pO(2) of 40 mmHg, a hypoxic limit of 5 mmHg, and a tissue/capillary radius ratio of 10 are used. For purely aerobic metabolism, a hypoxic fraction of 0.16 and volume-average pO(2) of 8 mmHg are calculated. Reducing the maximum oxygen rate constant by 9%, decreasing the tissue cylinder radius by 5%, or increasing the capillary radius by 8% abolishes the hypoxic fraction. When a glycolytic term is added, concentration profiles are similar to the base case. Using a distribution of tissue/capillary radius ratios increases the hypoxic fraction and reduces sensitivity to the oxygen consumption rate, compared to the case with a single tissue/capillary radius ratio. This model demonstrates that hypoxia is quite sensitive to metabolic rate and geometric factors. It also predicts quantitatively the effects of inhibited oxygen metabolism and enhanced mass transfer on tumor oxygenation.     

Hemoglobin-based O2 carrier O2 affinity and capillary inlet pO2 are important factors that influence O2 transport in a capillary

Hemopure (Biopure; Cambridge, MA) and PolyHeme (Northfield Laboratories; Evanston, IL) are two acellular hemoglobin-based O2 carriers (HBOCs) currently in phase III clinical trials for use as red blood cell substitutes. The most common adverse side effect that these HBOCs exhibit is increased vasoconstriction. Autoregulatory theory has been presented as a possible explanation for this physiological effect, where it is hypothesized that low-affinity HBOCs over-deliver O2 to tissues surrounding arterioles, thereby eliciting vasoconstriction. In this paper, we wanted to investigate HBOC oxygenation of tissue surrounding a capillary, which is the smallest element of the circulatory system. An a priori model has been developed in which the performance of mixtures of acellular HBOCs (synthesized by our group and others) and human red blood cells (hRBCs) has been simulated using a Krogh tissue cylinder model (KTCM) comprising a capillary surrounded by a capillary membrane and skeletal muscle tissue in cylindrical coordinates with specified tissue O2 consumption rates and Michaelis-Menten kinetics. In this study, the total hemoglobin (hRBCs and HBOCs) concentration was kept constant. The HBOCs studied possessed O2 affinities that were higher and lower compared to hRBCs (P50's spanned 5-55 mmHg), and the equilibrium binding/release of oxygen to/from the HBOCs was modeled using the Adair equation. At normoxic inlet pO2's, there was no correlation between O2 flux out of the capillary and the O2 affinity of the HBOC. However, a correlation was found between the average pO2 tension in the capillary and the O2 affinity of the HBOC. Additionally, we studied the change in the O2 equilibrium curve of HBOCs with different O2 affinities over a wide range of inlet pO2's and found that changing the inlet pO2 greatly affected which HBOC, having a unique O2 affinity, best delivered O2 to the surrounding tissue. The analysis of oxygen transport presented could lead to a better prediction of which acellular HBOC is best suited for a specific transfusion application that many times depends on the capillary inlet pO2 tension.     

Mathematical modeling of oxygen transport in skeletal muscle

Inhomogeneous perfusion of capillary beds can result in large-scale diffusion of oxygen between distant portions of an organ. The conceptual model of a single capillary supplying oxygen to a surrounding concentric cylinder of tissue is not applicable to a consideration of such processes. An entirely different approach to the modeling of oxygen transport to tissue, with specific reference to the capillary beds of skeletal muscle, is presented here. This approach is intended to replace the theoretical Krogh cylinder model of capillary-tissue oxygen transport with a much more realistic model that takes into account inhomogeneities of capillary density, blood flow velocity, and oxygen concentration inherent in the micro-vasculature. The oxygen distribution in inhomogeneously perfused skeletal muscle is analyzed mathematically by defining an averaged concentration profile that neglects the fine-scale variation from capillary to capillary.     

Local oxygen gradients near isolated mitochondria.

The oxygen concentration in tissue can vary on several length scales. The basic scale of variation is determined by capillary spacing. It is this scale that is manifest in the simplest Krogh cylinder model. A second, smaller scale of variation is associated with the consumption of oxygen by mitochondria. This paper gives a theoretical analysis of these smaller-scale oxygen variations near an isolated mitochondrion. To illustrate the effects of shape, we have carried out the calculations for prolate spheroids as well as for spheres. The principal result is that the local drop in oxygen pressure around a consuming mitochondrion is of the order of (gamma/3K) (3V/4 pi)2/3, where gamma is the oxygen consumption rate per unit mitochondrial volume, K is the Krogh oxygen diffusivity of the surrounding tissue, and V is the mitochondrial volume. The theory is applied to skeletal muscle in vivo and to hepatocytes in cell suspension experiments. In both cases, we find that local oxygen variations produced by oxygen consumption are much smaller than the cell-wide variations produced by the collective effect of all the mitochondria. For example, in maximally consuming skeletal muscle, the drop in oxygen pressure around a consuming mitochondrion is only of the order of 0.03 Torr.     

The effects of axial diffusion and permeability barriers on the transient response of tissue cylinders. I. Solution in transform space

Transient mass transfer in a Krogh tissue cylinder is described by a model taking into account axial diffusion in both blood and tissue, a localized permeability barrier at the capillary membrane and a diffusion barrier on the outer surface and at the ends of the cylinder. Radial diffusion in both blood and tissue is assumed to be infinitely fast. In contrast to previous work, which has usually relied on numerical methods for solving the equations, an exact solution is presented here in Laplace transform space. This allows calculation of the moments of the concentration at any point in the cylinder. Numerical results indicate that the moments of the residence time distribution are affected by the boundary conditions used, and that the discrepancies between the predictions using different conditions may be large in some physiological situations. Order-of-magnitude calculations are used to estimate when the use of simpler models may be feasible. The transform space solution may also be useful for parameter estimation, but it seems preferable to extend the present results to a time-domain solution for this purpose.     

Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios

We extend the validity of the quasi-steady state assumption for a model double intermediate enzyme-substrate reaction to include the case where the ratio of initial enzyme to substrate concentration is not necessarily small. Simple analytical solutions are obtained when the reaction rates and the initial substrate concentration satisfy a certain condition. These analytical solutions compare favourably with numerical solutions of the full system of differential equations describing the reaction. Experimental methods are suggested which might permit the application of the quasi-steady state assumption to reactions where it may not have been obviously applicable before.     

Event-driven time-optimal control for a class of discontinuous bioreactors

Discontinuous bioreactors may be further optimized for processing inhibitory substrates using a convenient fed-batch mode. To do so the filling rate must be controlled in such a way as to push the reaction rate to its maximum value, by increasing the substrate concentration just up to the point where inhibition begins. However, an exact optimal controller requires measuring several variables (e.g., substrate concentrations in the feed and in the tank) and also good model knowledge (e.g., yield and kinetic parameters), requirements rarely satisfied in real applications. An environmentally important case, that exemplifies all these handicaps, is toxicant wastewater treatment. There the lack of online practical pollutant sensors may allow unforeseen high shock loads to be fed to the bioreactor, causing biomass inhibition that slows down the treatment process and, in extreme cases, even renders the biological process useless. In this work an event-driven time-optimal control (ED-TOC) is proposed to circumvent these limitations. We show how to detect a "there is inhibition" event by using some computable function of the available measurements. This event drives the ED-TOC to stop the filling. Later, by detecting the symmetric event, "there is no inhibition," the ED-TOC may restart the filling. A fill-react cycling then maintains the process safely hovering near its maximum reaction rate, allowing a robust and practically time-optimal operation of the bioreactor. An experimental study case of a wastewater treatment process application is presented. There the dissolved oxygen concentration was used to detect the events needed to drive the controller.     

Analysis of the introduction and removal of glycerol in rabbit kidneys using a Krogh cylinder model

In 1982, Rubinsky and Cravalho described a Krogh cylinder model for the analysis of cryoprotectant transport in a perfused organ. By application of the Kedem-Katchalsky equations, changes in tissue volume caused by movements of water and solute were used to predict changes in capillary radius (Cryobiology 19, 70-82, 1982). We have now measured the changes in vascular resistance that are produced when sucrose or glycerol is introduced into the perfusate flowing through rabbit kidneys at 10 degrees C, and have analyzed these data by means of the Rubinsky-Cravalho semiempirical model. The sucrose data provided an estimate of hydraulic conductivity and the dimensions of the Krogh tissue units. Three rates of addition of glycerol, 10, 30, and 90 mM/min to a final concentration of 3 M, were studied. The vascular resistance fell to approximately 40% of its initial value (radius approximately 128% of initial value) with all three rates of addition, and then returned toward its normal value while the glycerol concentration was still increasing. This behavior could be explained either by a sudden change in solute permeability at that capillary radius, or by an inverse dependence of reflection coefficient upon solute concentration. Evidence is presented that favors the latter interpretation. The best fits for the apparent hydraulic conductivity and apparent solute permeability for glycerol are 1 X 10(-6) cm/sec atm and 6 X 10(-8) cm/sec, respectively, with the reflection coefficient falling from 1.0 when the glycerol concentration is zero to 0.1 when it is 3 M. The model is used to predict tissue concentrations of glycerol throughout each experiment.     

Capillary supply regions

The diffusion and consumption of substrate from capillaries are basic in human physiology. The general solution for the concentration in a region containing many parallel non-homogeneous capillaries is found. Except in very special cases, capillary supply regions cannot be approximated by Krogh's cylinder or Voronoi polygonal cylinders.     

Modeling of a Catalyzed Reaction Using Lipase Immobilized in a Poly(vinyl alcohol) Membrane

A mathematical model for the hydrolysis reaction of p‐nitro phenol laurate catalyzed by a lipase immobilized in a membrane was developed. In an earlier study this model reaction was found to show very different reaction rates when it was performed in aqueous micellar solution with free enzyme and with membrane immobilized enzyme. It was assumed that a local accumulation of substrate in the membrane is responsible for the observed rate enhancement. The conversion of p‐nitro phenol ester within the membrane was modeled by considering a combination of the convective flow through poly(vinyl alcohol) membrane pores, concentration polarization of substrate containing micelles at the membrane surface and the kinetics of the reaction with free enzymes. It was demonstrated that the model offered a comprehensive understanding of the interaction of the involved phenomena. The modeling results are in good agreement with the experimental data from 10 runs with different enzyme and substrate concentrations. The substrate concentration at the membrane surface increased by up to a factor of 3 compared to the feed concentration. This effect explains the observed rate enhancement. Moreover, the model was used to determine the unknown parameters, i.e., the intrinsic retention and the mass transfer coefficient, by fitting the model to the experimental data. The model may also be used to calculate the optimum operating conditions and design parameters of such a reactor.     

A material-balance approach for modeling bacterial chemotaxis to a consumable substrate in the capillary assay

A mathematical model was developed to simulate the movement of chemotactic bacteria into and within a capillary tube containing a metabolizable chemoattractant. The model was based on a material balance that accounts for the transport of bacteria and chemoattractant as well as consumption of the substrate throughout the capillary assay system. By solving the model with a numerical method, it was possible to predict the concentration of substrate and bacteria at points within the capillary and throughout the chamber. The model was tested for its ability to simulate the results of capillary assay experiments performed with Pseudomonas putida G7 and one of its chemoattractants, naphthalene, under conditions wherein naphthalene consumption was expected to affect the flux of bacteria into the capillary. While variations in the chemotactic responses were evident among different experiments, the model could simulate the accumulation of cells in the capillary using previously determined parameters, including the chemotactic sensitivity and random motility coefficients, chi(0) and mu. In particular, model predictions were consistent with the experimental observation that the chemotactic response in the capillary is diminished under conditions wherein consumption would be expected to be significant.     

A quantitative framework for the design of acellular hemoglobins as blood substitutes: implications of dynamic flow conditions

The delivery of oxygen to tissue by cell-free carriers eliminates intraluminal barriers associated with red blood cells. This is important in arterioles, since arteriolar tone controls capillary perfusion. We describe a mathematical model for O(2) transport by hemoglobin solutions and red blood cells flowing through arteriolar-sized tubes to optimize values of p50, Hill number, hemoglobin molecular diffusivity and concentration. Oxygen release is evaluated by including an extra-luminal resistance term to reflect tissue oxygen consumption. For low consumption (i.e., high resistance to O(2) release) a hemoglobin solution with p50=15 mmHg, n=1, D(HBO2)=3 x 10(-7) cm(2)/s delivers O(2) at a rate similar to that of red blood cells. For high consumption, the p50 must be decreased to 5 mmHg. The model predicts that regardless of size, hemoglobin solutions with higher p50 will present excess O(2) to arteriolar walls. Oversupply of O(2) to arteriolar walls may cause constriction and paradoxically reduced capillary perfusion.     

Fuzzy reasoning system for fault diagnosis of physiological activities in a cultivating process.

Aiming at development of a system which supports cultivating operations, a method to diagnose physiological activities in a cultivating process is presented, and a fuzzy expert system for diagnosing Lactobacillus casei cultivating process is implemented in this paper. This system can calculate specific rates of cell growth, substrate consumption, and product formation with measuring cell mass concentration, substrate concentration, and product concentration by using a turbidity sensor and HPLC. A database is implemented, where standard curves on specific rates representing characteristics of microorganisms are stored according to normalized substrate consumption. Comparing the calculated specific rates with standard values derived from the database, the system diagnoses physiological activities of the microorganisms. As a case study, a knowledge base for diagnosing lactic acid production process is implemented. The use of fault diagnosis on pH malfunctions by the expert system proves its reasonable performance.     

Oxygen balance for small organisms: an analytical model

An analytical model is developed that describes oxygen transport and oxygen consumption for small biological structures without a circulatory system. Oxygen inside the organism is transported by diffusion alone. Oxygen transfer towards the organism is retarded by a thin static fluid film at the surface of the organism. The thickness of this film models the outward water conditions, which may range from completely stagnant water conditions to so-called well-stirred water conditions. Oxygen consumption is concentration-independent above a specified threshold concentration (regulator behaviour) and is proportional to the oxygen concentration below this threshold (conformer behaviour). The model takes into account shape and size of the organism and predicts the transition from (pure) regulator behaviour to (pure) conformer behaviour, as well as the mean oxygen consumption rate. Thereby the model facilitates a proper analysis of the physical constraints set on shape and size of organisms without an active internal oxygen transport mechanism. This analysis is carried out in some detail for six characteristic shapes (infinite sheet, cylinder and beam; finite cylinder, sphere and block). In a well-stirred external medium, a flattened shape appears to be the most favourable for oxygen supply, while a compact shape (cube) is more favourable if the external medium is nearly stagnant. The theoretical framework is applied to oxygen consumption data of eight teleost embryos. This reveals relative insensitivity to external flow conditions in some species (e.g., winter flounder, herring), while others appear to rely on external stirring for a proper oxygen supply (e.g., largemouth bass). Interestingly, largemouth bass is the only species in our analysis that exhibits ‘fin-fanning’.     

Double site enzyme and squatting: where one regulatory ligand is also a substrate of the reaction

A regulatory model termed "squatting" has already been proposed (Mazat et al., 1977). It involves competition between at least two ligands binding to two regulatory sites. In this paper we study the case where one of the ligands is a substrate or a product of the reaction, one of the sites where competition occurs being the catalytic one. Complex regulatory patterns are evidenced and some experimental examples are analysed according to this model. It is worth noting that this model takes into account competition between chemically related ligands as occurs in vivo.     

A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand.

Oxygen transport from capillaries to exercising skeletal muscle is studied by use of a Krogh-type cylinder model. The goal is to predict oxygen consumption under conditions of high demand, on the basis of a consideration of transport processes occurring at the microvascular level. Effects of the decline in oxygen content of blood flowing along capillaries, intravascular resistance to oxygen diffusion, and myoglobin-facilitated diffusion are included. Parameter values are based on human skeletal muscle. The dependence of oxygen consumption on oxygen demand, perfusion, and capillary density are examined. When demand is moderate, the tissue is well oxygenated and consumption is slightly less than demand. When demand is high, capillary oxygen content declines rapidly with axial distance and radial oxygen transport is limited by diffusion resistance within the capillary and the tissue. Under these conditions, much of the tissue is hypoxic, consumption is substantially less than demand, and consumption is strongly dependent on capillary density. Predicted consumption rates are comparable with experimentally observed maximal rates of oxygen consumption.     

Near-field absorption in a circular cylinder from electric and magnetic line sources

A homogeneous, lossy circular cylinder is used as a simple model of a biological object in which interior heating is produced by the absorption of electromagnetic waves. For this model, we determined the optimum frequency, polarization, orientation and shape of applicators. Analytical and numerical results are given for both electric and magnetic line sources, with three different polarizations relative to the cylinder. Coupling efficiencies and contour plots are presented for a range of parameters. One particularly interesting result is the production of maximum energy deposition at the center of a cylinder of muscle tissue when exposed in the 100-MHz frequency range by the use of four applicators surrounding the cylinder.     

The method of Hill's coefficient calculation for the region of inhibition of an enzyme by excess of the substrate. Inhibition of phosphofructokinase by excess of ATP]

The new method for Hill's coefficient (nH) calculation in the region of substrate concentrations where the latter acts as an inhibitor has been developed. The method does not need preliminary determination of maximum value of enzyme reaction rate (V) for ascending branch of the plot of enzyme reaction rate versus substrate concentration and allows to avoid over-estimation of value of nH when the magnitude of optimal reaction rate is less than value of V. The literature data for inhibition of phosphofructokinase by excess of ATP are used for illustration of applicability of the suggested method of Hill's coefficient calculation.     

Continuous cultures limited by a gaseous substrate: Development of a simple, unstructured mathematical model and experimental verification with Methanobacterium thermoautotrophicum

This article presents a simple, unstructured mathematical model describing microbial growth in continuous culture limited by a gaseous substrate. The model predicts constant gas conversion rates and a decreasing biomass concentration with increasing dilution rate. It has been found that the parameters influencing growth are primarily the gas transfer rate and the dilution rate. Furthermore, it is shown that, for correct simulation of growth, the influence of gaseous substrate consumption on the effective gas flow through the system has to be taken into account.Continuous cultures of Methanobacterium thermoautotrophicum were performed at three different gassing rates. In addition to the measurement of the rates of biomass production, product formation, and substrate consumption, microbial heat dissipation was assessed using a reaction calorimeter. For the on-line measurement of the concentration of the growth-limiting substrate, H(2), a specially developed probe has been used. Experimental data from continuous cultures were in good agreement with the model simulations. An increase in gassing rate enhanced gaseous substrate consumption and methane production rates. However, the biomass yield as well as the specific conversion rates remained constant, irrespective of the gassing rate. It was found that growth performance in continuous culture limited by a gaseous substrate is substantially different from "classic" continuous culture in which the limiting substrate is provided by the liquid feed. In this report, the differences between both continuous culture systems are discussed.     

On facilitated oxygen diffusion in muscle tissues.

The role of myoglobin in facilitated diffusion of oxygen in muscle in examined in a tissue model that utilizes a central supplying capillary and a tissue cylinder concentric with the central capillary, and that includes the nonlinear characteristics of the oxygen-hemoglobin dissociation reaction. In contrast to previous work, this model exhibits the effect of blood flow and a realistic, though ideal, tissue-capillary geometry. Solutions of the model equations are obtained by a singular-perturbation technique, and numerical results are discussed for model parameters of physiologic interest. In contrast to the findings of Murray, Rubinow, Taylor, and others, fractional order perturbation terms obtained for the "boundary-layer" regions near the supplying capillaries are quite significant in the overall interpretation of the modeling results. Some closed solutions are found for special cases, and these are contrasted with the full singular-perturbation solution. Interpretations are given for parameters of physiologic interest.