Bipartite expressions for diffusional mass transport in biomembranes

Analytical expressions for solute diffusion through a membrane barrier for different initial and boundary conditions are available in the literature. The three commonest initial and boundary conditions are for a membrane without solute respectively immersed in a solution of constant concentration, immersed in such a solution for one side but with the other side isolated, and immersed in such a solution for one side and with the other side kept at zero concentration. The physical quantities for the first two initial and boundary conditions are concentration and average concentration (the total solute entering the membrane) with amperometric current (flux) and solute that permeates through the membrane (charge passed) for the third initial and boundary condition. Expressions for these methods in the literature are inconvenient for practical applications because of the infinite mathematical series required. An investigation of convergence of these expressions was therefore carried out. Simple but accurate bipartite expressions for these methods were constructed and provided theoretical support for studies on mass transport characterization of biomembranes. As a specific application, these expressions enabled a direct fit of the simulated observables to experimental values to obtain diffusion coefficients. For these initial and boundary conditions and corresponding physical quantities, simple one point methods for diffusion coefficient estimation are also suggested. These latter diffusion coefficients can be initial values for numerical fit methods.     

Ion transport and displacement currents with membrane-bound carriers

Summary The standard carrier model for ion transport by a one-to-one mechanism is developed to predict the time-dependent currents for systems that are symmetrical at zero applied potential. The complete solution for ions and carriers bearing any charge is derived by assuming that the concentration of ions in the membrane is low and either that the applied potential is small or that the applied potential affects equally all of the association and dissociation reactions between the ions and the carriers. The response to an abruptly applied potential is then given by the sum of a constant and two declining exponential terms. The time constants of these relaxations are described by the equations derived for neutral carriers by Stark, Ketterer, Benz and Läuger in 1971 (Biophys. J. 11:981). The sum of the amplitudes of the exponentials for small applied potentials obeys a relation like that first derived by Markin and Liberman in 1973 (Biofizika 18:453). For small applied potentials expressions are also provided for the voltage transients in charge-pulse experiments and for the membrane admittance.     

The kinetics of solute clearance by a single tissue capillary of limited permeability: Explicit solutions of two models and their bounds

To describe stationary-state kinetics of solute clearance from an interstitial space with an initial uniform concentration, an explicit solution of the Sangren-Sheppard model and a stratagem for an explicit solution of the Johnson-Wilson model are presented. In both cases expressions for computing the exact intravascular and extravascular concentrations at any time and location are complicated and inconvenient. Simpler and far more convenient formulae for determining upper and lower bounds on solute concentrations and on the pseudo-first-order clearance rate ‘constant’ (k) are derived. For the Johnson-Wilson model, the bounds are so tight thatk, for example, can be estimated with considerable accuracy whenever the capillary blood flow exceeds the permeability-surface area product.     

Mathematical analysis of oxygen concentration in a two dimensional array of capillaries

 An approach is presented for modeling transport and exchange in skeletal muscle that can be used to analyze vascular beds consisting of a large number of interacting capillaries. First the oxygen concentration is determined in a functional unit consisting of a single capillary surrounded by a region of tissue in which a flux is prescribed on the outer boundary of the region. This flux, which is a result of the interaction among all of the capillaries comprising the vascular bed, is then found by matching the concentration along the borders between adjacent units. This leads to a system of ordinary differential equations for the oxygen concentration in the capillaries coupled with a system of algebraic equations for the fluxes. The method is illustrated by obtaining the oxygen concentration within an array of capillaries for the case when each capillary has a different initial concentration and for the case when each capillary has a different flow rate. Received: 12 June 2001 / Revised version: 18 April 2002 / Published online: 17 January 2003 Key words or phrases: Skeletal muscle – Transport – Microcirculation     

A mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment

A situation is considered in which a fluid containing a substance flows through a vessel at a constant rate, the substance being permeable to the vessel wall. In the region outside the vessel there is supposed to be rapid mixing in the direction perpendicular to the axis of the vessel but no mixing longitudinally. The solution for the spatial distribution at any time is given for the case of an arbitrary initial distribution along the vessel length in the absence of an input. The solution is also given for the case of a single impulsive input, the concentration being initially zero everywhere. Work performed under Contract W-7405-eng-26 for the Atomic Energy Commission.     

Analytic solution to steady-state radial diffusion of a substrate with first-order reaction kinetics in the tissue of a Krogh's cylinder

It is often useful to calculate the concentration profile for a substrate undergoing reaction in the tissue surrounding a capillary. In this paper, we consider a model geometry consisting of a long straight cylinder of tissue surrounding a capillary. Substrate diffuses radially out of the capillary through the tissue, with consumption of substrate in the tissue directly proportional to substrate concentration (i.e., first-order reaction kinetics). The model is extended to include the case where a cylinder of necrotic tissue surrounds a metabolically active inner tissue cylinder. A simple analytic solution is derived, and concentration profiles are generated for various combinations of parameters. Compared to the case where substrate consumption is independent of concentration, this model predicts much more rapid depletion of substrate near the capillary interface. This can have significant implications for the calculation of the hypoxic fraction (e.g., tissue with pO(2)<0.5-5 mmHg) when tumor oxygenation is modeled. The model also permits calculation of the limiting substrate concentration for cell viability when the reaction rate constant is known and vice versa.     

Optimum capillary number for oxygen delivery to tissue in man

The optimum number of total capillaries in the whole human body was estimated from the analysis of the efficiency for oxygen (O₂) transport in the vascular-tissue system. We used a tissue model composed of uniform spheres in which O₂ diffuses from the capillary located at the centre of each sphere towards the surrounding tissue consuming O₂ at a constant rate. The tissue mass supplied by a single capillary was estimated as the area of positive O₂ concentration under a given condition of capillary flow and O₂ consumption rate. Total tissue mass was determined as the function of the capillary numbern and the total blood flow. On the other hand, the energy cost required to maintain the vascular system withn terminals was assessed by using the minimum volume model (Kamiya and Togawa,Bull. math. Biophys. 34, 431–438, 1972). The efficiency of the entire vascular-tissue system was evaluated by calculating the ratio of total tissue mass/cost function. The result of the calculation using physiological data of cardiac output and O₂ consumption for an average human adult during a heavy exercise revealed the maximum efficiency occurring at the capillary number 3.7×10¹⁰ which coincided well with its normal range of physiological estimates (3.2×10¹⁰–4.2×10¹⁰). We concluded that the entire vascular-tissue system is constructed so as to attain the highest efficiency in O₂ transport at its maximum activity.     

Some mathematical aspects of chemotherapy—II: The distribution of a drug in the body

In a previous paper (Bellman, Jacquez, and Kalaba,Bull. Math. Biophysics,22: 181–198, 1960) a model of the processes occurring in the exchange of a drug between capillary plasma, extracellular space and intracellular space was developed. This included the possibility of a reaction between the drug and a component of the intracellular space. The equations developed thus describe the events within a capillary bed. In the present paper, a simplified model of the body is set up. Each organ is treated as a single capillary bed and is linked to other organs via the circulation, in the parallel and/or series arrangements found in the body. Mixing in the circulation is included at the simplest possible level. The concentration of drug entering any one capillary bed is determined by the concentrations leaving all other capillary beds, the time lags, and mixing involved in the circulation. The equations describing these processes in conjunction with the equations of the processes occurring within each capillary bed lead to a large set of differential-difference equations.     

A singular dispersion relation arising in a caricature of a model for morphogenesis

In 1983 Oster et al. proposed a model for morphogenesis consisting of a system of partial differential equations in which the dispersion relation for the problem linearised about the zero solution has a singularity. That is, the initial growth rate of a small perturbation of wave number k from the zero solution tends to positive or negative infinity as k tends to some critical value k _c from above or below respectively. We consider here as a caricature of the model a single partial differential equation with a similar dispersion relation in a bounded one-dimensional domain. The wave number, or equivalently the domain size, may be thought of as a bifurcation parameter. For the Neumann problem a phenomenon arises in which, as the domain size l increases past a critical value l _l ,the linear stability of the n-th mode jumps from one solution to a remote solution. That is, for l _n the trivial solution is unstable and a certain non-trivial solution is stable to perturbations of mode n, whereas for l>l _n the opposite is true. For the Dirichlet or the Robin problem a linear stability change in the trivial solution occurs, but no corresponding change in any other solution has been found. The corresponding initial boundary value problems are then considered. An asymptotic analysis is performed in the weakly nonlinear limit in the particular case in which only one mode is unstable and gives an asymptotic solution for two classes of nonlinearity, one symmetric and the other asymmetric about u=0. A development of the method of harmonic balance is then used to obtain approximate solutions in the strongly nonlinear case and when more than one mode may be unstable.     

Toxic dinoflagellates in the Mediterranean sea

Abstract

A review of the distribution of toxic dinoflagellates and related toxic events in the Mediterranean Sea is provided. Diarrhetic shellfish poisoning (DSP) and paralytic shellfish poisoning (PSP) are the main seafood contaminations, that have been reported since 1987. In 1994 four lethal cases for PSP were reported in Morocco. DSP seems to be restricted in the Northern part of the basin, while PSP contamination presents a wider distribution. Potentially DSP species, belonging to the genus Dinophysis, are widely distributed in the Mediterranean Sea, suggesting a wider extension of this contamination. The lack of monitoring programs in the Southern Mediterranean could be responsible of DSP overlooking. The species responsible for PSP are Gymnodinium catenatum in the Alboran Sea and Alexandrium minutum in various parts of the basin. Ichtyotoxic effects, due to Gymnodinium and Gyrodinium species, have been sporadically reported in some Mediterranean areas.     

On the application of strain factors for approximation of the contribution of anisotropic cells to the mechanics of a tissue construct

Bio-artificial tissue constructs consisting of fibroblast cells embedded in a collagenous matrix are valuable in vitro systems in which to study cellular mechanics. Deriving cellular mechanics from the results of experimentation on tissue constructs requires a mathematical relationship that delineates amongst the contributions of the constituents of a tissue construct. A scaling between the average strain in a uniformly stretched tissue and the axial strain in isotropic cells was used in earlier work to study relations between cell mechanics and the overall mechanics of a tissue construct. That work showed that a scaling factor called a "strain factor" provided an accurate representation of the average axial strain in isotropic cells. The present study analyzes such relationships for anisotropic cells. We incorporate Eshelby's (1957; Proceedings of the Royal Society of London A 241, 376; 1959; Proceedings of the Royal Society of London A 252, 561) exact solution for the strain field in isolated ellipsoidal inclusions into the Zahalak (Biophysical journal 79, 2369) constitutive model for tissue constructs. Results showed that, for the case of prolate cells, the strain along the major cell axis is mostly influenced by the remote strain projected along that axis; off-axis cell mechanics plays only a small role in most tissues. The strain factor approximation is shown to be accurate for anisotropic cells to within a few percent for the vast majority of tissues. The results presented in this paper provide an explicit measure of the effects of cellular anisotropy, and a mechanism for calculating the contributions of these effects to overall tissue mechanics when these effects are important.     

Der O2-Austausch zwischen Capillaren und Gewebe

Summary The distribution of oxygen concentration or oxygen partial pressure (=po₂) in models with different capillary arrangements is calculated. The capillaries form homogeneously perfused concurrent and countercurrent systems or inhomogeneously perfused capillary mesh systems with different velocities of blood flow on the inside of the capillaries. The question arises as to what extent the distribution of the po₂ in the tissue and the capillaries depends on the differences in streaming velocities and directions of the blood. The results show, that the greatest gradients in the po₂ distribution are to be found in the capillary mesh system. A typical example with a set of constants after Thews (1960) shows that the differences between the venous po₂ and the lowest po₂ in the tissue is 32 mm Hg in the case of the capillary network, 9 mm Hg in the case of the concurrent and 3 mm Hg in that of the countercurrent system.     

Patient-Specific Artery Shrinkage and 3D Zero-Stress State in Multi-Component 3D FSI Models for Carotid Atherosclerotic Plaques Based on <i>In Vivo</i> MRI Data

Image-based computational models for atherosclerotic plaques have been developed to perform mechanical analysis to quantify critical flow and stress/strain conditions related to plaque rupture which often leads directly to heart attack or stroke. An important modeling issue is how to determine zero stress state from in vivo plaque geometries. This paper presents a method to quantify human carotid artery axial and inner circumferential shrinkages by using patient-specific ex vivo and in vivo MRI images. A shrink-stretch process based on patient-specific in vivo plaque morphology and shrinkage data was introduced to shrink the in vivo geometry first to find the zero-stress state (opening angle was ignored to reduce the complexity), and then stretch and pressurize to recover the in vivo plaque geometry with computed initial stress, strain, flow pressure and velocity conditions. Effects of the shrink-stretch process on plaque stress/strain distributions were demonstrated based on patient-specific data using 3D models with fluid-structure interactions (FSI). The average artery axial and inner circumferential shrinkages were 25% and 7.9%, respectively, based on a data set obtained from 10 patients. Maximum values of maximum principal stress and strain increased 349.8% and 249% respectively with 33% axial stretch. Influence of inner circumferential shrinkage (7.9%) was not very noticeable under 33% axial stretch, but became more noticeable under smaller axial stretch. Our results indicated that accurate knowledge of artery shrinkages and the shrink-stretch process will considerably improve the accuracy of computational predictions made based on results from those in vivo MRI-based FSI models.     

脂联素及其受体在乳腺癌中的表达及其与临床病理特征的相关性分析

目的:观察和分析脂联素(ADPN)及脂联素受体(adipo R)在乳腺癌中的表达及其与临床病理特征的相关性。方法:选取60例乳腺癌患者作为病例组,选取30例良性乳腺疾病患者作为对照组,对两组患者血清ADPN水平进行检测和比较。对病例组患者肿瘤组织及对照组患者病变组织中的ADPN、adipo R1、adipo R2表达水平进行检测和比较。对病例组患者的肿瘤原发部位、肿瘤结节数量、病理类型、肿瘤局部浸润情况、淋巴结转移情况、T分期、TNM分期情况及其与ADPN、adipo R1、adipo R2水平的关联性进行观察和分析。结果:病例组患者的血清ADPN水平及肿瘤乳腺组织中ADPN、adipo R1、adipo R2表达水平均显著低于对照组,两组之间的差异均有统计学意义(P0.05)。乳腺癌患者血清ADPN水平和乳腺组织中ADPN表达水平与肿瘤的局部浸润情况、T分期、淋巴结转移情况、TNM分期具有关联性(P0.05),乳腺癌患者乳腺组织中adipo R1和adipo R2表达水平与肿瘤的病理类型、局部浸润情况、T分期、淋巴结转移情况、TNM分期具有关联性(P0.05)。结论:乳腺癌患者外周血中的ADPN及肿瘤组织中的ADPN及其受体均呈现低表达,而且其表达水平与肿瘤的病理类型、浸润和转移情况及临床分期具有关联性,有望作为乳腺癌诊断和治疗的新型靶点。     

A Novel Zeolite Coating for Protection of Concrete Sewers from Biological Sulfuric Acid Attack

Microbial growth inhibition and resistance to biological deterioration of concrete specimens coated with silver-loaded zeolite was evaluated by measuring the time course of bacterial growth, biological sulfur oxidation, and sulfate production using Acidithiobacillus thiooxidans as a corrosive agent. Live bacterial cells declined from an initial inoculum concentration of 1.1 × 10₄ cell ml_-1 to zero in 10 days, during which only 0.5–1% of the initial sulfur concentration of 10 g l_-1 was biologically oxidized, corresponding to sulfate production rates of 35–42 mg SO ₄ ^{2 ?} g ^{? 1} S ^{? 1} . Leaching coefficients of calcium and silicon in the specimens coated with silver-loaded zeolite of 1.6 × 10 ^{? 4} to 4.6 × 10 ^{? 2} cm ² d ^{? 1} respectively, were only 0.8% and 1% of the uncoated specimens.     

Effect of turgor pressure on water permeability ofAllium cepa epidermis cell membranes

Summary Using onion epidermis layer a very accurate method for measuring the permeability of epidermis cells to water was standardized. In this method a 1.4 cm diameter epidermis disc was soaked in tritiated water (500 Ci/ml) for about 1 hr. Next the disc was mounted in a specially designed elution chamber where it was held flat and washed on the noncuticular side with ordinary water. A constant flow rate, high enough to minimize unstirred layer effect, was used. Permeability was calculated in the usual way after separating different exponentials from the efflux curve of tritiated water. Turgor pressure of the cell was regulated by soaking the disc in mannitol solutions containing tritiated water and washing it in the chamber with same concentration mannitol solution containing no radioactivity. Water permeability values were found to decrease less than 8% when the turgor pressure was decreased from 8 atm (full turgor) to zero. Turgor pressure had no significant effect on the water permeability of onion epidermal cells. Our results are contradictory to the findings of Zimmerman and Steudle (1974,J. Membrane Biol. 16:331) but are similar to the findings of Tazawa and Kamiya (1966,Aust. J. Biol. Sci. 19:339) and Kiyosawa and Tazawa (1972,Protoplasma 74:257).     

Physio-chemical mathematical theory for transitions in cellular metabolism: An aspect of carcinogenesis

It is pointed out that the non-monotonic character of the chemical reaction rate expressions, together with the relative magnitude of the diffusivity constants, is likely to engender a multiplicity of locally stable steady-state solutions to the system of reaction-diffusion equations for the concentration distributions of molecular species through the volume of a living cell. A transition in cellular metabolism, i.e., the dynamical evolution from an initial locally stable steady-state solution for the concentration distributions to another distinct locally stable steady-state solution, can be induced by an etiologic agent which modifies the rate expressions significantly during an interval of time. Global inequality analysis is employed to derive a condition on the modified rate expressions that is sufficient to guarantee the occurrence of such a transition in cellular metabolism. The possibility of a transition induced by a chemical carcinogen is investigated by applying the latter sufficient condition, and it is found that the statistical frequency of carcinogenesis should depend essentially on the magnitude of the grouping (T ^{2 − α} D ^α) for a total doseD of carcinogen administered to an animal at a uniform rate (D/T) over a time interval of durationT, where α is a certain positive number less than 1. This theoretical result is shown to be supported by the available experimental evidence.     

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.     

The Influence of Ambient Nitrate, Temperature, and Light on Nitrate Assimilation in Sudangrass Seedlings

Seedlings of Sundangrass (Sorghum Sudanese [Piper] Stapf.) were grown 10 to 13 days of age in a nutrient solution containing nitrate and then placed under treatment conditions for 24 h before assays of nitrate assimilation were begun. Nitrate uptake was determined by its disappearance from the ambient solution. In vivo reduction of nitrate was determined by the overall balance between the amount taken up and the change in tissue concentration of nitrate during the experiments. Nitrate reductase activity was determined from tissue slices. In vivo reduction was strongly regulated by uptake in response to time and ambient nitrate concentration, temperature and light. Nitrate reduction responded to the concentration of nitrate supplied by uptake and by a storage pool, since reduction often exceeded uptake. Nitrate reductase activity in tissue slices was exponential in initial response to increasing temperature. After a 24-h equilibration period at each temperature, the activity was lower at higher temperatures. In contrast, actual reduction of nitrate increased linearly with increasing temperature between 15 and 24°C in the plants equilibrated 24 h at each temperature. Nitrate uptake and reduction were greatly inhibited under low light conditions, with reduction inhibited more than uptake., The effect of ambient nitrate, temperature, and light on the nitrate assimilatory processes help to explain observations reported on nitrate accumulation by Sudangrass forage.     

Approximation method for electrolyte diffusion through a fixed charge medium

The diffusion of a binary salt through media containing uniformly dispersed immobile ions is formally equivalent to the diffusion of a nonelectrolyte with a concentration-dependent diffusion coefficient. This permits the application of a quasi-steady-state approximation, developed in a previous paper (Bull. Math. Biophysics,21, 19–32, 1959), to problems of transient diffusion. Free diffusion across an interface involving a discontinuity in immobile ion concentration is considered and expressions for the salt concentration and electrical potential at any point are developed. The dependence of electrical potential on the fixed charge density, ionic mobilities, as well as on the boundary concentrations is illustrated by a numerical example. This particular example is suggested by experiments on connective tissue.