Compartmental modeling of human lactation

A difficult task for a novice modeler is beginning a modeling project. The methods described here delineate the modeling process step by step. Aristole is quoted as having said "One learns to play the flute by playing the flute," and the same principle holds somewhat for modeling. An excellent means for understanding modeling is to work with example models, altering rate coefficients and model structure, to learn first-hand the mechanics of the process. An excellent resource for viewing and downloading published models is the Library of Mathematical Models of Biological Systems, and can be accessed via the World Wide Web at the address http://gopher.dml.georgetown.edu/model/model.html++ + . This modeling library provides access to working models of biological systems. Experimenting with available examples of models, such as those presented in this volume and those available through the Modeling Library, provides an excellent opportunity for new modelers to get started in compartmental modeling.

Computational studies of ion-water flux coupling in the airway epithelium. II. Role of specific transport mechanisms

Ion and water balance by the in vivo airway epithelium was investigated utilizing dynamic computer modeling. Parameters of the osmotically significant transport processes were varied to assess the sensitivity of water transport and fluid composition to transport perturbations. Establishment and regulation of water secretion represent a coordinated function of at least seven different ion transport processes: basolateral passive potassium transport, basolateral active sodium-potassium transport, basolateral sodium-potassium-chloride cotransport, apical passive sodium and chloride transport, and diffusion of sodium and chloride across the paracellular path. We found that apical chloride permeability at a level reported for cystic fibrosis is sufficient to cause the airway dehydration characteristic of cystic fibrosis. Given the reduction in apical chloride permeability in cystic fibrosis, a reduction in apical sodium permeability can potentially compensate completely for the airway dehydration associated with the cystic fibrosis genetic defect. Other simulations presented here address the importance of various membrane transport processes in airway epithelium water balance and the sensitivity of epithelium water balance to ion transport perturbations.

Computational studies of ion-water flux coupling in the airway epithelium. I. Construction of model

A mathematical model of ion and water transport across the airway epithelium is presented. The model consists of 12 state variables representing ion concentrations, volumes, and membrane potentials. All osmotically significant membrane transport processes for which there is conclusive experimental evidence are included: passive apical sodium and chloride movement, basolateral sodium-potassium pumping, basolateral sodium-potassium-chloride cotransport, passive basolateral potassium movement, nonselective passive paracellular ion motion, and water transport across all membranes. Ion movements are described by Michaelis-Menten kinetics or by the constant field flux equation. Model parameters are established with Ussing chamber data. Model behavior is validated by comparing in vitro simulations with experimental results. The model accurately reproduces short-circuit chloride and sodium fluxes, short-circuit current, and open-circuit membrane potentials from Ussing chamber data in the secreting and nonsecreting states. The model is then used to describe the behavior of the airway epithelium in vivo, in which case the apical electrolyte compartment is small and of variable size and ionic composition.

Compartmental analysis of the dynamics of beta-carotene metabolism in an adult volunteer

Metabolism of a 73 mumol oral dose of beta-carotene-d8 in olive oil was determined from plasma beta-carotene-d8 and retinol-d4 concentration-time curves in an adult male. beta-Carotene-d8 and retinol-d4 concentrations in serial plasma were measured using high performance liquid chromatography (HPLC) and gas chromatography-mass spectrometry (GC-MS), respectively. Plasma beta-carotene-d8 and retinol-d4 concentration-time curves were described by a 5-term and a 3-term polyexponential equation, respectively, using an empirical description of beta-carotene metabolism. A physiologic compartmental model of beta-carotene metabolism was also constructed and tested. This model suggests that 22% of the beta-carotene dose is absorbed: 17.8% as intact beta-carotene and 4.2% as retinoid. Also, it suggests that both liver and enterocyte are important in converting beta-carotene to retinoid; 43% is converted in liver and 57% in enterocyte. Finally, it suggests that the mean residence time for beta-carotene is 51 days and that the 73 mumole dose does not alter the fractional transfer coefficients of the system after absorption takes place. The issue of central versus eccentric cleavage of beta-carotene in humans can be studied with further modeling combined with use of appropriately labeled beta-carotene.

Effect of lipid on inert gas kinetics

A Monte Carlo simulation of inert gas transit through skeletal muscle has been extended to include regions of increased gas solubility to simulate regions of high lipid content. Position of the regions within the simulation module was varied, as was the muscle-lipid partition coefficient (lambda). The volume percentage of the lipid regions (alpha) was varied from 0 to 25% while lambda covered the range from 1 to 50. The effects of alpha and lambda on mean transit time and on relative dispersion (RD; ratio of SD to the mean) were examined for a single lipid volume and compared with expected values under the assumption that the tissue is composed of two well-stirred compartments. Mean transit times varied from approximately 0.80 to 1.20 times the values predicted by a simple parallel two-compartment model, whereas RD varied from 0.9 to 3.6. For fixed lambda, RD as a function of lipid fraction passes through a maximum that is shifted and was also smaller than expected from a simple two-compartment model. For fixed alpha, RD approaches an asymptotic value for large lambda, but the asymptote is smaller than that expected from the two-compartment model. When lipid is distributed in only two regions, RD decreases with increasing separation of the regions and with increasing surface area of the fat regions. A model of two well-stirred compartments that allows mixing between the compartments yields results similar to those from the simulation.

Contribution of tissue lipid to long xenon residence times in muscle

Experiments demonstrate that the mean residence time of an inert gas in tissue is longer than that predicted by a single-compartment model of gas exchange. Also the relative dispersion (RD, the standard deviation of residence times divided by the mean) is 1 according to this model, but RDs in real tissues are closer to 2, suggesting that a multiple-compartment model might be more accurate. The residence time of a gas is proportional to its solubility in the tissue. Although the noble gases in particular are 10 times more soluble in lipid than in nonlipid tissues, models of gas exchange generally do not incorporate measurements of the lipid in tissue, which may lead to error in the predicted gas residence times. Could a multiple-compartment model that accounts for the lipid in tissue more accurately predict the mean and RD of gas residence times? In this study, we determined the mean and RD of Xe residence times in intact and surgically isolated muscles in a canine model. We then determined the lipid content and the perfusion heterogeneity in each tissue, and we used these measurements with a multiple-compartment model of gas exchange to predict the longest physiologically plausible Xe residence times. Even so, we found the observed Xe mean residence times to be twice as long as those predicted by the model. However, the predicted RDs were considerably larger than the observed RDs. We conclude that lipid alone cannot account for the residence times of Xe in tissue and that a multiple-compartment model is not an accurate representation of inert gas exchange in tissue.(ABSTRACT TRUNCATED AT 250 WORDS)

Quantifying the effect of intravascular perfluorocarbon on xenon elimination from canine muscle

Intravenous infusions of perfluorocarbon (PFC) may improve decompression sickness outcome in animals by accelerating inert gas elimination from tissue, but any such effect has not been quantified experimentally. In this study we used an animal model of tissue Xe kinetics to test this hypothesis and to quantify the effect of PFC. Eight dogs were ventilated with dilute 133Xe in air for 4 h of Xe uptake. Four dogs were then given an infusion (20 ml/kg iv) of a 40% (vol/vol) perfluorodecalin-glycerol emulsion, and four control dogs were given only isotonic glycerol. All were then switched to open-circuit air breathing for 4 h of Xe elimination. During this time Xe radioactivity-time curves were recorded from two intact hind leg muscles, and the Xe mean residence times during elimination were estimated using an analysis by moments and compared by group. Tissue blood flows were measured using microspheres once during Xe uptake and twice during Xe elimination, and cardiac outputs were measured by thermodilution at 30-min intervals. In the PFC group the measured circulating PFC fraction increased the calculated Xe solubility by an average factor of 1.77 and so was expected to increase the Xe elimination rate by 77%. The observed Xe mean residence times on elimination for the PFC group averaged 33.5 min [95% confidence interval (CI) 19.5-47.6] compared with the glycerol control average of 70.1 min (95% CI 56.1-84.2), representing an increase in the rate of Xe elimination by a factor of 2.09 or 109%.(ABSTRACT TRUNCATED AT 250 WORDS)

The nature of ion and water barrier crossings in a simulated ion channel

Using a combination of techniques, including molecular dynamics, time-correlation analysis, stochastic dynamics, and fitting of continuum diffusion theory to electrophysiological data, a characterization is made of thermally driven sodium, water, and D2O motion within the gramicidin A channel. Since the channel contents are constrained to move in a single-file fashion, the motion that corresponds to experimentally measurable rates of permeation of the membrane is the motion of the center of mass of the channel contents. We therefore emphasize channel contents center-of-mass motion in our analysis of molecular dynamics computations. The usual free energy calculation techniques would be of questionable validity when applied to such motion. As an alternative to those techniques, we postulate a periodic sinusoidal free energy profile (related to the periodic structure of the helical channel) and deduce the fluid dynamic diffusion coefficient and the height and spacing of the free energy barriers from the form of the mean-square-deviation function, using stochastic computations. The fluid dynamic friction in each case appears similar to that for aqueous solution. However, the diffusive motions are modulated by a spatially periodic free energy profile with a periodicity characteristic of an L-D pair of amino acids in the gramicidin helix, approximately 1.7 A in the model we use. The barrier height depends on which substance is moving in the channel, but in each case is several times thermal energy. For barriers of this width and height, the motion is intermediate between the low-friction (transition-state) and high-friction (Brownian) limits. Thus, neither of these formalisms that have been used commonly to describe membrane permeation gives an accurate picture of the underlying physical process (although the Brownian description seems closer to correct). The non-Markovian Langevin equation must be solved to describe properly the statistics of the process. The "channel state of matter" characteristic of the channel contents appears to have some properties typical of the solid and some typical of the liquid state. The magnitude of the local friction and nature of the ion solvation are similar to the liquid state, but the periodicities of structure, free energy, and dynamics are somewhat solid-like. The alignment of water dipoles in the channel bears some resemblance to the orientational ordering of a nematic liquid crystal, but unlike a nematic liquid crystal, the waters have a degree of translational order as well. Thus, the "channel state" is not adequately described by analogy to either the solid or liquid states or to liquid crystals but must be dealt with as its own characteristic type of condensed matter.

Xenon kinetics in muscle are not explained by a model of parallel perfusion-limited compartments

Experimental tissue gas kinetics do not follow the prediction for a single stirred perfusion-limited compartment. One hypothesis proposes that the kinetics might be explained by considering the tissue as a collection of parallel compartments, each with its own flow, reflecting the tissue microcirculatory flow heterogeneity. In this study, observed tissue gas kinetics were compared with the kinetics predicted by a model of multiple parallel compartments. Gas exchange curves were generated by recording the time course of tissue radioactivity in the intact calf muscles of anesthetized ventilated dogs exposed to step function changes of 133Xe in the inspired air for 5-h periods. Microcirculatory flow heterogeneity in the same tissue was determined by the radioactive microsphere method. Observed mean tissue transit times were on average longer than predicted by a factor of 6.7. Observed means averaged 52.1 min compared with 8.3 min predicted by the perfusion-limited model. Relative dispersions of tissue transit times were also uniformly larger than predicted. We conclude that Xe gas kinetics in intact canine skeletal muscle are not explained by a model of multiple parallel perfusion-limited compartments. Countercurrent exchange of gas between vessels is a possible explanation.