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.      

The Kinetics of Sodium Transport in the Toad Bladder : I. Determination of the transport pool

A compartmental model of toad bladder sodium content has been developed, whereby it is possible to measure the four unidirectional fluxes across the opposite faces of the transport compartment, as well as the amount of sodium in the compartment. ²⁴Na is added to the mucosal medium of a short-circuited bladder mounted between halves of a chamber in which the fluid is stirred by rotating impellers. After a steady state is reached, nonradioactive medium is flushed through both sides of the chamber, collected, and counted. The data from each chamber are fitted to sums of exponentials and interpreted in terms of conventional compartmental analysis. Three exponentials are required, with half-times of 0.2, 2.2, and 14.0 min. It is shown that the first of these represents chamber washout, the second the transport pool, and the third a tissue compartment which is not involved in active sodium transport and which does not communicate with the transport pool. The second compartment contains 10.5 µEq of sodium per 100 mg dry weight, an amount equal to approximately 30% of total tissue sodium. The results also indicate, as expected from electrophysiological data, that the mucosal-facing side of the transport compartment is over 10 times as permeable to sodium as the serosal, or pump, side.     

Contributions to the treatment of tracer distribution equations in multicompartmental systems

Summary A tracer kinetic method is outlined for the treatment of multicompartmental non steady-state systems. Metabolic rates are considered as consisting of two components: one of them (exchange) is balanced by a reverse process of the same rate, and the other (transport) results in concentration change. Assuming the transport rates to be known, the method allows of excluding a lot of logically possible compartmental models as inconsistent with the experimental data. Thus the real compartment structure of the system can be approached.     

The elementary mass action rate constants of P-gp transport for a confluent monolayer of MDCKII-hMDR1 cells

The human multi-drug resistance membrane transporter, P-glycoprotein, or P-gp, has been extensively studied due to its importance to human health and disease. Thus far, the kinetic analysis of P-gp transport has been limited to steady-state Michaelis-Menten approaches or to compartmental models, neither of which can prove molecular mechanisms. Determination of the elementary kinetic rate constants of transport will be essential to understanding how P-gp works. The experimental system we use is a confluent monolayer of MDCKII-hMDR1 cells that overexpress P-gp. It is a physiologically relevant model system, and transport is measured without biochemical manipulations of P-gp. The Michaelis-Menten mass action reaction is used to model P-gp transport. Without imposing the steady-state assumptions, this reaction depends upon several parameters that must be simultaneously fitted. An exhaustive fitting of transport data to find all possible parameter vectors that best fit the data was accomplished with a reasonable computation time using a hierarchical algorithm. For three P-gp substrates (amprenavir, loperamide, and quinidine), we have successfully fitted the elementary rate constants, i.e., drug association to P-gp from the apical membrane inner monolayer, drug dissociation back into the apical membrane inner monolayer, and drug efflux from P-gp into the apical chamber, as well as the density of efflux active P-gp. All three drugs had overlapping ranges for the efflux active P-gp, which was a benchmark for the validity of the fitting process. One novel finding was that the association to P-gp appears to be rate-limited solely by drug lateral diffusion within the inner monolayer of the plasma membrane for all three drugs. This would be expected if P-gp structure were open to the lipids of the apical membrane inner monolayer, as has been suggested by recent structural studies. The fitted kinetic parameters show how P-gp efflux of a wide range of xenobiotics has been maximized.     

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.     

Kinetics of glucose transport and sequestration in lactating bovine mammary glands measured in vivo with a paired indicator/nutrient dilution technique.

To quantify kinetics of mammary glucose utilization in vivo, 24 paired glucose and extracellular indicator (p-aminohippuric acid) dilution curves across intact bovine mammary glands were obtained after bolus injections into the external iliac artery. Dilution curves were analyzed using a compartmental capillary, convolution integration model. Four candidate submodels of glucose transport and metabolism in capillary supply zones were fit to the glucose dilution curves and evaluated. Model I, with one extracellular compartment for glucose and first-order unidirectional uptake, failed, indicating that efflux of glucose from the intracellular space could not be ignored. Model II, with first-order exchanges between extracellular and intracellular compartments and sequestration from the latter, was overdefined because unidirectional clearance of glucose was at least five times the blood flow rate and 20 times the net clearance rate. Model III, combining extracellular and intracellular space into one compartment, was superior in its goodness-of-fit to curves and identifiability of parameters. Michaelis-Menten parameters of sequestration were not identifiable. Parameters of the optimal compartmental capillary, convolution integration model were applicable to both the dynamics of injected glucose dilution and the steady-state background arteriovenous difference of glucose. Glucose sequestration followed first-order kinetics between 0 and 7 mM extracellular glucose with an average rate constant of 0.006 s(-1) or a clearance of 44 ml/s. The ratio of intracellular to extracellular glucose distribution space was 0.34, which is considerably lower than the expected intracellular volume and suggests an intracellular occlusion compartment with which extracellular glucose rapidly exchanges.     

Multi-compartmental modelling and experimental design for glucose transport studies in insulin-resistant rats

Many pathologies are associated with abnormalities of glucose metabolism or with perturbations of its transport (type 2 diabetes or insulin-resistance). The pre-diabetic state is characterised by a state of insulin-resistance, in others words a defect of glucose transport in insulin-sensible tissues, such as muscles and adipose tissues. The mathematical modelling of experimental data can be an excellent method to explore the mechanisms implied in the studied biological phenomenon. Thus, starting from a symbolic formulation like the compartmental modelling, it can be possible to develop a theoretical basis for the observation and to consider the best-adapted experiments for the study. We showed with mathematical models that [123I]-6-deoxy-6-iodo-D-glucose (6-DIG), shown as a tracer of glucose transport in vitro, could point out this transport abnormality. To quantify the insulin resistance, we estimated the fractional transfer coefficients of 6-DIG from the blood to the organs. We realised many studies to lead to a satisfying model; special attention has been paid to the precision of the parameter to select the best model. The results showed that by associating experimental data obtained with 6-DIG activities and an adapted mathematical model, discriminating parameters (in and out fractional transfer coefficients) between the two groups (control and insulin-resistant rats) could be pointed out.     

The Kinetics of Sodium Transport in the Toad Bladder : II. Dual effects of vasopressin

In the accompanying paper, a compartmental model for the toad bladder sodium transport system was developed. In the present paper, the model is tested by determining the effects of antidiuretic hormone on the pools and fluxes. It is shown that this hormone affects only that sodium pool previously designated as the transport pool, and that the effects are on two separate sites. In the first place, the hormone stimulates entry at the mucosal side of the transport compartment, and by this means brings about an increase in the amount of sodium contained in the compartment. Second, the hormone has a distinct stimulatory effect on the rate coefficient for efflux across the serosal boundary, the pump rate coefficient. Evidence is presented that under control conditions, the pump rate coefficient is a decreasing function of the pool size, a characteristic feature of a saturating system. Therefore, the effect of vasopressin in increasing both the pool size and the pump rate coefficient must be construed as a direct effect on the pump, and not one which is secondary to the increase in the pool size. Furthermore, it is shown that the effect of the hormone on the sodium pump is not dependent on the presence of sodium in the serosal medium.     

A compartmental model of magnesium metabolism in healthy men based on two stable isotope tracers

The aim of this study was to build a compartmental model of magnesium (Mg) kinetics by using data collected from six healthy adult men after oral administration of 26Mg and intravenous administration of 25Mg. Blood, urine, and feces were collected for 12 days after administration of the isotopes. Isotopic ratios were determined by inductively coupled plasma-mass spectrometry. Data were analyzed for each subject using SAAMII. We began with a compartmental model previously proposed (Avioli LV and Berman M. J Appl Physiol 21: 1688-1694, 1966) and developed an alternative approach to resolve the discrepancy between model-predicted curves and experimental data. This analysis enables the exploration of 25% of total body Mg that exchanges rapidly from plasma compartment with two extraplasma pools. One of the extraplasma compartments contains 80% of the exchangeable Mg with a transport rate of 48 +/- 13 mg/h. The second exchanges 179 +/- 88 mg of Mg/h. The model permitted estimation of kinetic parameters as well as fractional Mg absorption and fecal endogenous excretion.     

On stochastic compartmental modeling

This communication contains a proof of the fact that the coefficient of variation of the contents of a compartment of a stochastic compartmental model with deterministic rate parameters is small for large populations. We can therefore conclude that the use of stochastic compartmental models is not of great consequence in the case of systems involving large populations when only the randomness of the transfer mechanism is considered.     

Role of the Paracellular Pathway in Isotonic Fluid Movement Across the Renal Tubule

Evidence for a highly permeable paracellular shunt in the proximal tubule is reviewed. The paracellular pathway is described as a crucial site for the regulation of net absorption and for solute-solvent interaction. Available models for the coupling of salt and water transport are assessed with respect to the problem of isotonic water movement. Two new models are proposed taking into account that the tight junctions are permeable to salt and water and that active transport sites for sodium are distributed uniformly along the lateral cell membrane. The first model (continuous model) is a modification of Diamond and Bossert''s proposal using different assumptions and boundary conditions. No appreciable standing gradients are predicted by this model. The second model (compartmental model) is an expansion of Curran''s double membrane model by including additional compartments and driving forces. Both models predict a reabsorbate which is not isosmotic. For the particular case of the proximal tubule it is shown that in the presence of a leaky epithelium these deviations from isotonicity might have escaped experimental observation.     

Kinetic parameters in the compartmental analysis of lithium transport in Lemna gibba using the stable isotopes, 6Li and 7Li, as tracers

Due to the lack of a convenient radiotracer of Li, we have used the stable isotopes, ⁶Li and ⁷Li, for unidirectional flux measurements in Lemna gibba L. (GI). Detection was performed using an ionic microanalyser. The conventional method of compartmental analysis has been improved in several ways. Growth of the plant samples has been taken into consideration. The possible non-uniqueness of the kinetic solution fitting the experimental points has been considered by numerical calculation. The derivatives and not only the direct functions have been used in order to detect whether a constant term had to be added to the first-order terms. The validity of the estimates of the characteristic parameters of transport has been checked by using efflux and influx data simultaneously. The "manipulation-shock" effects have been shown not to be negligible. The kinetic data have no simple biological significance per se, but they can serve to calculate cellular parameters of transport when combined with appropriate stereometric data. These considerations are relevant not only to our present problem, but to any type of compartmental analysis, whatever the plant sample and the absorbed substrate.     

Myocardial blood perfusion and transport modeling using inert-tracer techniques: a review and recent investigations

The inert-gas clearance method for measuring blood perfusion in the heart may be useful in detecting and assessing coronary disease and myocardial infarctions. Estimating perfusion from clearance data requires a model of tracer transport. The tracer transport models in use are the compartmental model, the kinetic model, and more complex models which yield estimates by optimal estimation techniques. The implementation of one such complex model in which tissue need not be assumed homogeneous, and the resulting myocardial perfusion and diffusibility estimates, are discussed. Methods are reviewed which may be used to detect and assess coronary disease by average and regional myocardial-perfusion measurements. Possible explanations for the observed multicompartment myocardial clearance curve are discussed.     

Extension of compartmental parameters to blocks of compartments with application to lipoprotein kinetics

Suppose that the compartments of a compartmental model are separated into blocks (sets of compartments). In general, the blocks can not be regarded as compartments but it may be possible to construct a “condensation model,” the compartments of which correspond to the blocks, in such fashion so as to retain certain salient properties of the blocks. Condensation is a way of formally summarizing a large model by presenting a smaller one to emphasize certain characteristics of the larger model. Suppose that the parameters to be retained are the mean residence times through the blocks; this paper deals with the construction of the condensation model, the properties of the condensation model, and the possible applications of condensation to data analysis, particularly in regard to lipoprotein kinetics.     

Continuous Monitoring of Fluxes of Photoassimilate in Leaves and Whole Plants

A technique is described for continuous, simultaneous and invivo monitoring of the photosynthetic uptake of carbon, thetransport of carbon from the leaf, and its transport throughoutthe entire plant. Continuous but varying amounts of carbon-11 dioxide are suppliedto a leaf while observing the amount of labelled photoassimilatethat appears in different parts of the plant. Appropriate analysisof these tracer profiles gives physiologically meaningful parameterssuch as the fraction of recent assimilate which is exportedfrom the leaf. Variation in these parameters can be followedover an indefinite time, to follow the effects of natural rhythmsor treatments. The method does not require any assumptions aboutmechanism, nor the assumption that transport processes withinthe leaf are constant, unlike compartmental analysis of effluxdata sometimes used to infer fluxes in leaves. The method is illustrated by an experiment with a young barleyplant where the export fraction from a leaf is found to reducethrough the day, and to respond immediately after the leaf isshaded. Key words: Partitioning, ¹¹C, compartmental analysis     

Kinetic modeling of [(18)F]FDG in skeletal muscle by PET: a four-compartment five-rate-constant model

Various modeling strategies have been developed to convert regional [(18)F]fluorodeoxyglucose ([(18)F]FDG) concentration measured by positron emission tomography (PET) to a measurement of physiological parameters. However, all the proposed models have been developed and tested mostly for brain studies. The purpose of the present study is to select the most accurate model for describing [(18)F]FDG kinetics in human skeletal muscle. The database consists of basal and hyperinsulinemic-euglycemic studies performed in normal subjects. PET data were first analyzed by an input-output modeling technique (often called spectral analysis). These results provided guidelines for developing a compartmental model. A new model with four compartments and five rate constants (5K model) emerged as the best. By accounting for plasma and extracellular and intracellular kinetics, this model allows, for the first time, PET assessment of the individual steps of [(18)F]FDG kinetics in human skeletal muscle, from plasma to extracellular space to transmembrane transport into the cell to intracellular phosphorylation. Insulin is shown to affect transport and phosphorylation but not extracellular kinetics, with the transport step becoming the main site of control. The 5K model also allows definition of the domain of validity of the classic three-compartment three- or four-rate-constant models. These models are candidates for an investigative tool to quantitatively assess insulin control on individual metabolic steps in human muscle in normal and physiopathological states.     

Indistinguishability and identifiability analysis of linear compartmental models.

Two compartmental model structures are said to be indistinguishable if they have the same input-output properties. In cases in which available a priori information is not sufficient to specify a unique compartmental model structure, indistinguishable model structures may have to be generated and their attributes examined for relevance. An algorithm is developed that, for a given compartmental model, investigates the complete set of models with the same number of compartments and the same input-output structure as the original model, applies geometrical rules necessary for indistinguishable models, and test models meeting the geometrical criteria for equality of transfer functions. Identifiability is also checked in the algorithm. The software consists of three programs. Program 1 determines the number of locally identifiable parameters. Program 2 applies several geometrical rules that eliminate many (generally most) of the candidate models. Program 3 checks the equality between system transfer functions of the original model and models being tested. Ranks of Jacobian matrices and submatrices and other criteria are used to check patterns of moment invariants and local identifiability. Structural controllability and structural observability are checked throughout the programs. The approach was successfully used to corroborate results from examples investigated by others.     

Capillaries within compartments: microvascular interpretation of dynamic positron emission tomography data

Measurement of exchange of substances between blood and tissue has been a long-lasting challenge to physiologists, and considerable theoretical and experimental accomplishments were achieved before the development of the positron emission tomography (PET). Today, when modeling data from modern PET scanners, little use is made of earlier microvascular research in the compartmental models, which have become the standard model by which the vast majority of dynamic PET data are analysed. However, modern PET scanners provide data with a sufficient temporal resolution and good counting statistics to allow estimation of parameters in models with more physiological realism. We explore the standard compartmental model and find that incorporation of blood flow leads to paradoxes, such as kinetic rate constants being time-dependent, and tracers being cleared from a capillary faster than they can be supplied by blood flow. The inability of the standard model to incorporate blood flow consequently raises a need for models that include more physiology, and we develop microvascular models which remove the inconsistencies. The microvascular models can be regarded as a revision of the input function. Whereas the standard model uses the organ inlet concentration as the concentration throughout the vascular compartment, we consider models that make use of spatial averaging of the concentrations in the capillary volume, which is what the PET scanner actually registers. The microvascular models are developed for both single- and multi-capillary systems and include effects of non-exchanging vessels. They are suitable for analysing dynamic PET data from any capillary bed using either intravascular or diffusible tracers, in terms of physiological parameters which include regional blood flow.     

Exploration of the intercellular heterogeneity of topotecan uptake into human breast cancer cells through compartmental modelling

A mathematical multi-cell model for the in vitro kinetics of the anti-cancer agent topotecan (TPT) following administration into a culture medium containing a population of human breast cancer cells (MCF-7 cell line) is described. This non-linear compartmental model is an extension of an earlier single-cell type model and has been validated using experimental data obtained using two-photon laser scanning microscopy (TPLSM). A structural identifiability analysis is performed prior to parameter estimation to test whether the unknown parameters within the model are uniquely determined by the model outputs. The full model has 43 compartments, with 107 unknown parameters, and it was found that the structural identifiability result could not be established even when using the latest version of the symbolic computation software Mathematica. However, by assuming that a priori knowledge is available for certain parameters, it was possible to reduce the number of parameters to 81, and it was found that this (Stage Two) model was globally (uniquely) structurally identifiable. The identifiability analysis demonstrated how valuable symbolic computation is in this context, as the analysis is far too lengthy and difficult to be performed by hand.