A study of nutritional transport in capillary-tissue exchange system

A two region flow model has been developed for a capillary-tissue exchange unit. Nutrient dissolved in plasma enters into the tissue from capillary through diffusion, filtration and osmosis. The governing non-linear coupled partial differential equations in the two regions (capillary and tissue) have been solved separately with suitable boundary and matching conditions. The results for the variation of Taylor's Diffusivity Coefficient and concentration of a nutrient in the tissue region have been brought out for various values of the parameters involved in the analysis and discussed. It has been particularly noted that the penetration depth in the tissue for the nutritional transport can be considered as an important diagnostic parameter for many cardiovascular diseases.     

On morphometric measurement of oxygen diffusing capacity in middle ear gas exchange.

An accurate mathematical model of transmucosal gas exchange is prerequisite to understanding middle ear (ME) physiology. Current models require experimentally measured gas species time constants for all extant conditions as input parameters. However, studies on pulmonary gas exchange have shown that a morphometric model that incorporates more fundamental physiochemical and anatomic parameters accurately simulates transport from which the species time constants can be derived for all extant conditions. Here, we implemented a variant of that model for ME gas exchange that requires the measurement of diffusional length (tau) for the ME mucosa. That measure contributes to the mucosal diffusing capacity and reflects the resistance to gas flow between air space and capillary. Two methods for measuring tau have been proposed: linear distance between the air-mucosal boundary and capillary and the harmonic mean of all contributing pathway lengths. Oxygen diffusing capacity was calculated for different ME mucosal geometries by using the two tau measures, and the results were compared with those predicted by a detailed, two-dimensional finite element analysis. Predictive accuracy was improved by incorporating the harmonic tau measure, which captures important information regarding variations in capillary shape and distribution. However, compared with the oxygen diffusing capacity derived from the finite element analysis, both measures yielded nonlinear, positively biased estimates. The morphometric techniques underestimate diffusion length by failing to account for the curvilinear gas flow pathways predicted by the finite element model.     

Effects of vasomotion and venous pressure elevation on capillary-tissue fluid exchange across hetero-porous membrane

Flow autoregulation in the arteriolar network serves to maintain the capillary-tissue fluid balance by regulation of capillary pressure. In the present study, we have examined theoretically the effects of arteriolar vasomotion and venous pressure elevation on the capillary fluid exchange, the interstitial fluid pressure, and the interstitial osmotic pressure during capillary pressure regulation. We used Starling's hypothesis and extended it to include a consideration of a parallel hetero-porous pathway and to determine the effects of plasma protein filtration on interstitial fluid pressure and osmotic pressure. We have found that arteriolar vasomotion plays a primary role in protecting the capillary-tissue fluid balance during the elevation of capillary flow and that it is a secondary mechanism for the regulation of capillary arterial pressure.     

Mathematical studies of capillary-tissue exchange

A mathematical model of capillary-tissue exchange is presented and the method of solution of the resulting equations is described. The model includes the mutual interaction of fluid movement across the capillary wall and the convection and diffusion of a number of solutes. A variety of solutions for situations of physiological interest are obtained and discussed.     

Assessment of transcapillary glucose exchange in human skeletal muscle and adipose tissue

We studied the kinetics of glucose exchange between plasma and interstitial fluid (ISF) in human skeletal muscle and adipose tissue under fasting conditions. Five normal human subjects received an intravenous [6,6-2H2]glucose infusion in a prime-continuous fashion. During the tracer infusion, the open-flow microperfusion technique was employed to frequently sample ISF from quadriceps muscle and subcutaneous adipose tissue. The tracer glucose kinetics observed in muscle and adipose tissue ISF were found to be well described by a capillary-tissue exchange model. As a measure of transcapillary glucose exchange efficiency, the 95% equilibrium time was calculated from the identified model parameters. This time constant was similar for skeletal muscle and adipose tissue (28.6 +/- 3.2 vs. 26.8 +/- 3.6 min; P = 0.60). Furthermore, we found that the (total) interstitial glucose concentration was significantly lower (P < 0.01) in muscle (3.32 +/- 0.46 mmol/l) and adipose tissue (3.51 +/- 0.17 mmol/l) compared with arterialized plasma levels (5.56 +/- 0.13 mmol/l). Thus the observed gradients and dynamic relationships between plasma and ISF glucose in muscle and adipose tissue provide evidence that transcapillary exchange of glucose is limited in these two tissues under fasting conditions.     

The variance of the distribution of traversal times in a capillary bed

A random walk model of capillary tracer transit times is developed that treats simulataneously: plug flow in the capillary, radial and axial diffusion in the capillary cylinder and tissue annulus, and endothelial barriers to solute transport. The mean transit time is simply the volume of distribution divided by blood flow. Variance of transit times has additive terms for radial, axial, and barrier influences that are reduceable to variances of simpler models of capillary exchange. The dependence of variance on the solute diffusion coefficient is not monotonic, but has a minimum near 0·5 × 10?6 cm²/s for reasonable parameters and no barrier, Small molecules like inert gases are expected to have larger variances with higher diffusion coefficients, while larger molecules and barrier limited solutes will have the reverse dependence. Available literature data indicates that capillary heterogeneity will have a major influence on whole-body variance of transit times.     

Tracer measurements of non-equilibrium volumes of distribution

According to conventional theory the product of the transport flowrate and the mean transit time of a tracer through a system yields the equilibrium volume of distribution for the tracer, regardless of tracer kinetics or space geometry. Experimental results do not support this notion. The influence of measurement time on the volume measured with a bolus technique is addressed using systems theory to analyze a tissue-impedance form of the Sangren-Sheppard model. Assymptotic solutions show that the volume estimates are governed by a time constant, tau, related to diffusion in the tissue, to tissue capacity, and to wall permeability, and by a dimensionless ratio, f, describing a relation of tau to vascular transport time. A third parameter, g, describing the relative contributions to overall resistance to diffusion of effective permeability and of limited diffusivity in the tissue, is shown to be of less importance. The derived tau is similar to but not equivalent to the often cited "characteristic time". The "equilibrium" volume of distribution is defined as that which would be measured if equilibrium were allowed to establish. The "non-equilibrium" volume of distribution is defined as that which would be measured given finite times and is shown to approach the "equilibrium" volume as such times increase. Tracer equilibration is not required to accurately measure the "equilibrium" volume. When there is no flow limitation (f much less than 1) a measurement time of tau (plus vascular transit time) would yield a "non-equilibrium" volume only 33% of the "equilibrium" volume; a time of 2 tau would yield 55%; a time of 10 tau would yield effectively the total equilibrium volume. Finite diffusivity in tissue and permeability restrictions can have significant effects on the proportion of the volume measured.     

A geometrical model of dermal capillary clearance

A new microscopic model is developed to describe the dermal capillary clearance process of skin permeants. The physiological structure is represented in terms of a doubly periodic array of absorbing capillaries. Convection-dominated transport in the blood flow within the capillaries is coupled with interstitial diffusion, the latter process being quantified via a slender-body-theory approach. Convection across the capillary wall and in the interstitial phase is treated as a perturbation which may be added to the diffusive transport. The model accounts for the finite permeability of the capillary wall as well as for the geometry of the capillary array, based on realistic values of physiological parameters. Calculated dermal concentration profiles for permeants having the size and lipophilicity of salicylic acid and glucose illustrate the power and general applicability of the model. Furthermore, validation of the model with published in vivo experimental results pertaining to human skin permeation of hydrocortisone is presented. The model offers the possibility for in-depth theoretical understanding and prediction of subsurface drug distribution in the human skin following topical application, as well as rates of capillary clearance into the systemic circulation. A simpler approach that treats the capillary bed as a homogeneously absorbing zone is also employed. The latter may be used in conjunction with the capillary exchange model to estimate measurable dermal transport and clearance parameters in a straightforward manner.     

Mathematical models for exchange of substances in regional vascular beds— Measurement of the rates within vivo counters

Sangren and Sheppard developed a mathematical model for first-order processes taking place in the regional circulation, applicable—for example—to tracer studies of potassium transport. It permits calculation of specific activity at any point along a “tube of flow” or in the cuff of tissue surrounding it as a function of time following a spike injection of tracer. In efforts to relate to the exchange a rate curves obtained within vivo counters pointed at the region of interest, we developed a compartment-system model of the process. In investigating the properties of the Sangren and Sheppard model integrated over an entire circulatory bed, as thein vivo counter would see it, we found that when the distribution of transit times of the “tubes of flow” can be approximated by an exponential sum, the solution reduces to that of the compartment system model. This results in an important simplification in the calculation, and insight into the assumptions underlying the two different models. A curve-fitting computer program for the compartment model has been written and applied to double-isotope studies of potassium transport in the hind leg of the dog.     

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.     

Plasmodium coatneyi: alterations of transcapillary escape rate and capillary permeability to fibrinogen in rhesus monkeys

The transcapillary escape rate and plasma clearance of fibrinogen were studied in six rhesus monkeys infected with Plasmodium coatneyi as well as in six control monkeys using 131I-fibrinogen as a tracer. The mean transcapillary escape rate of fibrinogen in the infected group was significantly higher than that of the control group. Both plasma volume and plasma fibrinogen concentration were also elevated in the infected group, these resulting in a significantly higher intravascular fibrinogen mass, plasma clearance rate, and outflux of fibrinogen from the intravascular to the extravascular compartments. Both effective capillary pore area/unit path length available for restricted diffusion and the specific permeability coefficient of plasma fibrinogen in the infected monkeys were also found to be significantly higher than those of the control group. These findings indicated that there was an increased leakage of plasma fibrinogen from the circulation into the extravascular space which was due either to increased capillary surface area and/or to an increased capillary permeability in rhesus monkeys infected with P. coatneyi.     

Multibreath tracer species dynamics in the lung

By studying the behavior of various tracer species in the lungs, one can assess many important characteristics which distinguish normal and abnormal function. Quantitative evaluation of function depends on the use of an appropriate model in conjunction with experimental data. A multi-compartment model is derived from mass balances to describe dynamic as well as (breath-averaged) steady-state transport processes between the environment and pulmonary capillary blood. The breathing cycle is divided into three time periods (inspiration, expiration, and pause) so that the model equations are discrete in time. No other model of tracer species transport in the lungs deals simultaneously with species dynamics, variable breathing pattern, distribution inhomogeneities, and non-equilibrium between alveolar gas and capillary blood. Models currently in the literature are shown to be special cases of the model presented here.     

Peripheral-layer viscosity and microstructural effects on the capillary-tissue fluid exchange

A theoretical investigation of capillary-tissue fluid exchange has been studied including the characteristics and influence of the boundaries and media through which the fluid flows. Filtration from a cylindrical capillary into the concentrically surrounding tissue space and flow from a capillary into the tissue across a thin membrane are analyzed in detail. In has been observed that the filtration efficiency of the functional unit decreases as the viscosity of the peripheral layer increases. Contrary to the results of Apelblat [17], the slip velocity at the porous boundary plays a dominant role in filtration efficiency. It has also been noticed that the filtration efficiency decreases as the slip velocity at the porous boundary increases.     

Analytical model for long-distance tracer-transport in plants

Recent investigations of long-distance transport in plants using non-invasive tracer techniques such as ¹¹C radiolabeling monitored by positron emission tomography (PET) combined with magnetic resonance imaging (MRI) revealed the need of dedicated methods to allow a quantitative data analysis and comparison of such experiments. A mechanistic compartmental tracer transport model is presented, defined by a linear system of partial differential equations (PDEs). This model simplifies the complexity of axial transport and lateral exchanges in the transport pathways of plants (e.g. the phloem) by simulating transport and reversible exchange within three compartments using just a few parameters which are considered to be constant in space and time. For this system of PDEs an analytical solution in Fourier-space was found allowing a fast and numerically precise evaluation. From the steady-state behavior of the model, the system loss (steadily fixed tracer along the transport conduits) was derived as an additional parameter that can be readily interpreted in a physiological way. The presented framework allows the model to be fitted to spatio-temporal tracer profiles including error and sensitivity analysis of the estimated parameters. This is demonstrated for PET data sets obtained from radish, sugar beet and maize plants.     

Effects of red cell permeability on transcapillary tracer transport: The case of negligible back diffusion

Estimates of capillary tracer permeability calculated using multiple indicator data depend upon the particular model adopted to describe blood tissue exchange. The model proposed by Crone (1963) is appropriate when some of the injected tracer diffuses into the tissue but does not return appreciably to the bloodstream before data collection is terminated. Under these conditions extraction of tracer by the tissue depends on a single dimensionless parameter, α_cap, defined as the ratio of capillary permeability surface area to water flow. The effects of finite red cell tracer permeability on the Crone model estimate of capillary permeability are examined in the present study. The results indicate that even when back diffusion from the extravascular space is negligible, significant errors in the Crone model estimate can be expected when capillary permeability is relatively high and the ratio of red cell to capillary permeability is less than unity. However, when an aliquot of blood is equilibrated with tracer prior to injection and the dimensionless capillary permeability is relatively low (i.e. α_cap ≦ 0.25 for a haematocrit≦50%), the whole blood Crone model estimate of α_cap will be within 10% of the actual value, irrespective of red cell permeability. Red cell-plasma exchange for commonly used tracer-organ combinations should not significantly affect Crone estimates of capillary permeability under normal physiological conditions, but may be important in low flow situations. Supported by Grant No. HL 19153 (SCOR in Pulmonary Vascular Diseases). This work was done during Dr. Roselli’s tenure as an NHLBI Training Grant Fellow (NHLBI Training Grant No. 07123).     

Modeling kinetics of infused 13NN-saline in acute lung injury.

A mathematical model was developed to estimate right-to-left shunt (Fs) and the volume of distribution of 13NN in alveolar gas (VA) and shunt tissue (Vs). The data obtained from this model are complementary to, and obtained simultaneously with, pulmonary functional positron emission tomography (PET). The model describes 13NN kinetics in four compartments: central mixing volume, gas-exchanging lung, shunting compartment, and systemic recirculation. To validate the model, five normal prone (NP) and six surfactant-depleted sheep in the supine (LS) and prone (LP) positions were studied under general anesthesia. A central venous bolus of 13NN-labeled saline was injected at the onset of apnea as PET imaging and arterial 13NN sampling were initiated. The model fit the tracer kinetics well (mean r2 = 0.93). Monte Carlo simulations showed that parameters could be accurately identified in the presence of expected experimental noise. Fs derived from the model correlated well with shunt estimates derived from O2 blood concentrations and from PET images. Fs was higher for LS (54 +/- 18%) than for LP (5 +/- 4%) and NP (1 +/- 1%, P < 0.01). VA, as a fraction of PET-measured lung gas volume, was lower for LS (0.18 +/- 0.09) than for LP (0.96 +/- 0.28, P < 0.01), whereas Vs, as a fraction of PET-measured lung tissue volume, was higher for LS (0.46 +/- 0.26) than for LP (0.05 +/- 0.08, P < 0.01). The main conclusions are as follows: 1) the model accurately describes measured arterial 13NN kinetics and provides estimates of Fs, and 2) in this animal model of acute lung injury, the fraction of available gas volume participating in gas exchange is reduced in the supine position.     

Modelling advection and diffusion of water isotopologues in leaves

We described advection and diffusion of water isotopologues in leaves in the non-steady state, applied specifically to amphistomatous leaves. This explains the isotopic enrichment of leaf water from the xylem to the mesophyll, and we showed how it relates to earlier models of leaf water enrichment in non-steady state. The effective length or tortuosity factor of isotopologue movement in leaves is unknown and, therefore, is a fitted parameter in the model. We compared the advection-diffusion model to previously published data sets for Lupinus angustifolius and Eucalyptus globulus. Night-time stomatal conductance was not measured in either data set and is therefore another fitted parameter. The model compared very well with the observations of bulk mesophyll water during the whole diel cycle. It compared well with the enrichment at the evaporative sites during the day but showed some deviations at night for E. globulus. It became clear from our analysis that night-time stomatal conductance should be measured in the future and that the temperature dependence of the tracer diffusivities should be accounted for. However, varying mesophyll water volume did not seem critical for obtaining a good prediction of leaf water enrichment, at least in our data sets. In addition, observations of single diurnal cycles do not seem to constrain the effective length that relates to the tortuosity of the water path in the mesophyll. Finally, we showed when simpler models of leaf water enrichment were suitable for applications of leaf water isotopes once weighted with the appropriate gas exchange flux. We showed that taking an unsuitable leaf water enrichment model could lead to large biases when cumulated over only 1 day.     

Revision of the theory of tracer transport and the convolution model of dynamic contrast enhanced magnetic resonance imaging

Counterexamples are used to motivate the revision of the established theory of tracer transport. Then dynamic contrast enhanced magnetic resonance imaging in particular is conceptualized in terms of a fully distributed convection–diffusion model from which a widely used convolution model is derived using, alternatively, compartmental discretizations or semigroup theory. On this basis, applications and limitations of the convolution model are identified. For instance, it is proved that perfusion and tissue exchange states cannot be identified on the basis of a single convolution equation alone. Yet under certain assumptions, particularly that flux is purely convective at the boundary of a tissue region, physiological parameters such as mean transit time, effective volume fraction, and volumetric flow rate per unit tissue volume can be deduced from the kernel.      

A computational model of hemodynamic parameters in cortical capillary networks

The analysis of hemodynamic parameters and functional reactivity of cerebral capillaries is still controversial. To assess the hemodynamic parameters in the cortical capillary network, a generic model was created using 2D voronoi tessellation in which each edge represents a capillary segment. This method is capable of creating an appropriate generic model of cerebral capillary network relating to each part of the brain cortex because the geometric model is able to vary the capillary density. The modeling presented here is based on morphometric parameters extracted from physiological data of the human cortex. The pertinent hemodynamic parameters were obtained by numerical simulation based on effective blood viscosity as a function of hematocrit and microvessel diameter, phase separation and plasma skimming effects. The hemodynamic parameters of capillary networks with two different densities (consistent with the variation of the morphometric data in the human cortical capillary network) were analyzed. The results show pertinent hemodynamic parameters for each model. The heterogeneity (coefficient variation) and the mean value of hematocrits, flow rates and velocities of the both network models were specified. The distributions of blood flow throughout the both models seem to confirm the hypothesis in which all capillaries in a cortical network are recruited at rest (normal condition). The results also demonstrate a discrepancy of the network resistance between two models, which are derived from the difference in the number density of capillary segments between the models.     

Theoretical Analysis of Net Tracer Flux Due to Volume Circulation in a Membrane with Pores of Different Sizes : Relation to solute drag model

When osmotic pressure across an artificial membrane, produced by a permeable electrically neutral solute on one side of it, is balanced by an external pressure difference so that there is no net volume flow across the membrane, it has been found that there will be a net flux of a second electrically neutral tracer solute, present at equal concentrations on either side of the membrane, in the direction that the "osmotic" solute diffuses. This has been ascribed to solute-solute interaction or drag between the tracer and the osmotic solutes. An alternative model, presented here, considers the membrane to have pores of different sizes. Under general assumptions, this "heteroporous" model will account for both the direction of net tracer flux and the observed linear dependence of unidirectional tracer fluxes on the concentration of the osmotic solute. The expressions for the fluxes of solutes and solvent are mathematically identical under the two models. An inequality is derived which must be valid if the solute interaction model and/or the heteroporous model can account for the data. If the inequality does not hold, then the heteroporous model alone cannot explain the data. It was found that the inequality holds for most published observations except when dextran is the osmotic solute.