Parameter identification in N-compartment mamillary models

We consider a general mamillary model with a central compartment (compartment 1) and n?1 peripheral compartments, each bidirectionally connected to the first. Elimination is allowed from any compartment and effectively occurs from the system. With input introduced into an arbitrary compartment and measurement performed in an arbitrary compartment, explicit equations are given to derive the parameters of the model from the input-output procedure. The calculations include essentially the determination of the roots of a polynomial plus some elementary algebra. If input and measurement are performed in the same compartment, then a set of 2n elementary combinations of the model parameters can be uniquely determined. However, the model parameters themselves can only be localized, each within an interval. These intervals are explicitly calculated and their width discussed.     

On transfer times in tracer experiments

A compartment is defined as a pool of material whose behavior can be described by a deterministic or by a stochastic equation; these two equations are used to define the transit time through the compartment, the total residence time, the time of entrance and the time of exit.If in a complex system one or more compartments are accessible, the transport of material through it can be studied using a tracer. Then the transfer time between any two compartments, or through the cycle around a compartment, can be analyzed under certain hypotheses, even if the transport along the route considered cannot be described by compartment equations.     

Continous administration of short-lived radioisotope tracers and the analogous Laplace transform.

Short-lived radioactive tracers are used because of the low radiation dose to patients. Another advantage finding increasing use, however, is that the equilibrium activities achieved by continuous administration to a steady state contain kinetic information. This is not the case with long-lived isotopes.The derivation of quantitative kinetic information in the form of rate constants or flows requires the formulation of a model of the system being studied. Several approaches to this have been published based on a model of single compartments with simultaneous arrival of tracer. To deal with more realistic models a method is proposed which uses the analogy between the procedure of continuous administration of short-lived tracer and the Laplace transform. If f(t) is the activity of a long-lived tracer in any part of a system after administration of unit activity to the input, then

$\text{∫}\text{o}\text{∞}\text{e}{}^{\mathrm{?st}}\text{f(t)}\text{d}\text{t=}\text{f}\text{(s)}$

is the “equilibrium” activity of the part after continuous administration of a short-lived tracer of decay constant $\text{f}\text{(f)}$ is also a value of the Laplace transform of (t).This analogy permits all the theorems of Laplace transform theory to be applied to the analysis of measured activities. The basis of the analogy is explained and examples are given of its application to a number of models which represent actual physiology more realistically than single compartment models. In these applications the transformed equations representing the model, with measured values of activity inserted for each transform, are solved to derive the rate constants. This is different from the use of Laplace transforms where the constant coefficients are known and the initial value problem is solved to find the behaviour of the variables.     

Myosin-Powered Membrane Compartment Drives Cytoplasmic Streaming,Cell Expansion and Plant Development

Using genetic approaches, particle image velocimetry and an inert tracer of cytoplasmic streaming, we have made a mechanistic connection between the motor proteins (myosins XI), cargo transported by these motors (distinct endomembrane compartment defined by membrane-anchored MyoB receptors) and the process of cytoplasmic streaming in plant cells. It is shown that the MyoB compartment in Nicotiana benthamiana is highly dynamic moving with the mean velocity of ~3 μm/sec. In contrast, Golgi, mitochondria, peroxisomes, carrier vesicles and a cytosol flow tracer share distinct velocity profile with mean velocities of 0.6–1.5 μm/sec. Dominant negative inhibition of the myosins XI or MyoB receptors using overexpression of the N. benthamiana myosin cargo-binding domain or MyoB myosin-binding domain, respectively, resulted in velocity reduction for not only the MyoB compartment, but also each of the tested organelles, vesicles and cytoplasmic streaming. Furthermore, the extents of this reduction were similar for each of these compartments suggesting that MyoB compartment plays primary role in cytosol dynamics. Using gene knockout analysis in Arabidopsis thaliana, it is demonstrated that inactivation of MyoB1-4 results in reduced velocity of mitochondria implying slower cytoplasmic streaming. It is also shown that myosins XI and MyoB receptors genetically interact to contribute to cell expansion, plant growth, morphogenesis and proper onset of flowering. These results support a model according to which myosin-dependent, MyoB receptor-mediated transport of a specialized membrane compartment that is conserved in all land plants drives cytoplasmic streaming that carries organelles and vesicles and facilitates cell growth and plant development.     

On washout in nonlinear compartmental systems

Let x(t) be a solution of a compartmental system. If, for some compartment j, x_j(t)→0 as t→∞, then we say that the compartment j washes out. We show that a compartment washes out if it always reaches (along a fixed path) either the environment or another compartment for which there is no return path. Additional criteria, particularly regarding exponential convergence, are also presented. Examples are drawn from tracer kinetics, enzyme reactions, and epidemic models.     

On the distribution of the general irreversiblen-compartmental model having time-dependent transition probabilities

The cumulant generating function and first two moments are derived for the stochastic distribution of units in a general irreversiblen-compartment model with time-dependent transition probabilities. In this model, a unit in the first compartment can transfer to any one of the remainingn−1 compartments and a unit in the second compartment can transfer to any of the remainingn−2 compartments and so on. In addition, a unit can enter or leave the system through any compartment. The work is related to previous research and a numerical example is given.     

Stochastic theory of compartments: An open,two compartment,reversible system with independent poisson arrivals

This paper discusses two compartment models with interaction allowed between the compartments. The total number of particles in the system at any time is discussed along with the number to the found in each separate compartment. An interesting result is that the number of particles in each of the two compartments areindependent random variables. Some asymptotic results are also given. The paper is a continuation of some earlier work by the author.     

Measuring and Overcoming Limits of the Saffman-Delbrück Model for Soap Film Viscosities

We observe tracer particles diffusing in soap films to measure the two-dimensional (2D) viscous properties of the films. Saffman-Delbrück type models relate the single-particle diffusivity to parameters of the film (such as thickness h) for thin films, but the relation breaks down for thicker films. Notably, the diffusivity is faster than expected for thicker films, with the crossover at h/d = 5.2 ± 0.9 using the tracer particle diameter d. This indicates a crossover from purely 2D diffusion to diffusion that is more three-dimensional. We demonstrate that measuring the correlations of particle pairs as a function of their separation overcomes the limitations of the Saffman-Delbrück model and allows one to measure the viscosity of a soap film for any thickness.     

Cellular sodium transport in arterial wall - summary of studies by digital computer simulation

The distribution of electrolytes between extracellular and intracellular compartments is of greater functional significance in biological tissues than the total amount. A method commonly employed for the study of a substance distribution among different spaces is the continuous outflow and radioisotope counting of a fragment of tissue previously incubated with a radioactive tracet (here ²²Na⁺). The experimental data can be analyzed within the frame of a compartmental analysis system which is defined by transport rate constants (k_ij) and solved when all k_ij are numerically determined.In practical applications the direct solution requires measurements in different compartments or knowledge of the initial condition for each compartment, but this is not possible in most biological tissues where the experimentally observable data represent the summation of tracer in all compartments at each point in time. The availability of the SAAM (Simulation, Analysis And Modelling) computer program allows for a powerful simulation procedure to arrive at values for k_ij. Experimental data and specifications for a chosen model are entered in the SAAM format. A simulated ‘inflow’ experiment is mde by the computer by assigning to k_ij some estimate values. All initial conditions are now zero, since at the beginning there is no radioactive tracer in any compartment. After a time equivalent in the experimental realm to achieving constant specific activity, a ‘time change’ programmed into the computer takes place so that the outflow part of the experiment is developed with the same k_ij as for the inflow part, the final conditions for the inflow before the time change being the initial conditions for the outflow. An iterative procedure modifies the initial estimates of k_ij to seek the best fit.By coupling a powerful simulation procedure to an advanced radionuclide tracer biological methodology, we have been able to find effects of the hormone aldosterone and of experimental hypertension on Na⁺ distribution in arterial wall years before such effects were demonstrated by purely conventional experimental techniques. This computer simulation procedure may be suited to other experimental situations in which outflow data are the only available.     

Disposition of intraperitoneally injected calcium-45 in suckling rats

The time course of the concentration of radiocalcium was studied in the serum, skeleton, pelt, muscles, and pooled internal organs of 10-day-old rats. Within 10 hours of injection, the specific activity of the tissue groups exceeded the specific activity of the serum and remained above it during the period studied (120 hours). Chemical and autoradiographic analyses showed how rapidly most of the injected Ca⁴⁵ found its way into the skeleton. A model was constructed with the assumption that the skeleton constitutes an essentially irreversible reservoir for the tracer in a multicompartment system in which the blood is the central or feeding compartment. The rate of transfer of the tracer from the soft tissue compartments to the serum was calculated from the equation See PDF for Equation in which C = concentration in serum (expressed as a series of exponential terms) C' = concentration in a given soft tissue Substitution in the integrated form of this equation yielded equations which had the major properties of the empirical equations fitted to the experimental points. The relative order of transfer constants (k'_–1) was: organs ≥ pelt > muscle.     

Relationship between loading dose and infusion rate to achieve any fraction of the steady-state level in any compartment model

When a drug is infused at a constant rate K₀, the time necessary for the concentration to reach a satisfying threshold of effectiveness may be too long. To achieve this level faster, it is useful to give simultaneously a dose D, of the same drug by intravenous injection. This paper proposes the calculation, as a function of K₀ and model parameters, of the loading dose D necessary to reach, in a time T, any fraction of the asymptotic value of the amount of drug in a compartment receiving a constant rate infusion, for any n-compartment model. As an example, the expression of D for mammillary and catenary pharmacokinetic models is derived.     

Versatile analysis of single-molecule tracking data by comprehensive testing against Monte Carlo simulations

We propose here an approach for the analysis of single-molecule trajectories which is based on a comprehensive comparison of an experimental data set with multiple Monte Carlo simulations of the diffusion process. It allows quantitative data analysis, particularly whenever analytical treatment of a model is infeasible. Simulations are performed on a discrete parameter space and compared with the experimental results by a nonparametric statistical test. The method provides a matrix of p-values that assess the probability for having observed the experimental data at each setting of the model parameters. We show the testing approach for three typical situations observed in the cellular plasma membrane: i), free Brownian motion of the tracer, ii), hop diffusion of the tracer in a periodic meshwork of squares, and iii), transient binding of the tracer to slowly diffusing structures. By plotting the p-value as a function of the model parameters, one can easily identify the most consistent parameter settings but also recover mutual dependencies and ambiguities which are difficult to determine by standard fitting routines. Finally, we used the test to reanalyze previous data obtained on the diffusion of the glycosylphosphatidylinositol-protein CD59 in the plasma membrane of the human T24 cell line.     

Kinetic Modeling of 52Fe/52mMn-Citrate at tthe Blood-Brain Barrier by Positron Emission Tomography

The kinetics of iron at the blood-brain barrier of the monkey were studied in vivo using positron emission tomography (PET) and the tracer 52Fe/52mMn-citrate. 52mMn is the beta(+)-emitting daughter nuclide of 52Fe and therefore contributes to the observed signal and background in the PET images and may influence the quantification of physiological relevant iron parameters. The kinetics of pure (52m)Mn-citrate at the blood-brain barrier of the monkey were studied experimentally, and the analysis of the data with a reasonable compartment model led to equal efflux and influx parameters for Mn (1.35 +/- 0.3 x 10(-2) min(-1)). By using complexes between Mn and diethylenetriaminepentaacetic acid, the validity of the proposed model could be confirmed. To describe the observed kinetics of 52Fe/(52m)Mn-citrate, the manganese model was coupled to an iron model, which finally allowed the quantification of two iron-specific parameters: an input rate into global brain tissue of 7.15 +/- 2.6 x 10(-4) min(-1) and a time delay of roughly 24 min to account for the observed activities. The simpler linearization procedure has been proposed and could be applied to all our data sets and is able to replace the complicated nonlinear iron/manganese tracer kinetic model neglecting any influence of manganese on the analysis.     

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.     

Thematic review series: patient-oriented research. Design and analysis of lipoprotein tracer kinetics studies in humans

Lipoprotein tracer kinetics studies have for many years provided new and important knowledge of the metabolism of lipoproteins. Our understanding of kinetics defects in lipoprotein metabolism has resulted from the use of tracer kinetics studies and mathematical modeling. This review discusses all aspects of the performance of kinetics studies, including the development of hypotheses, experimental design, statistical considerations, tracer administration and sampling schedule, and the development of compartmental models for the interpretation of tracer data. In addition to providing insight into new metabolic pathways, such models provide quantitative information on the effect of interventions on lipoprotein metabolism. Compartment models are useful tools to describe experimental data but can also be used to aid in experimental design and hypothesis generation. The SAAM II program provides an easy-to-use interface with which to develop and test compartmental models against experimental models. The development of a model requires that certain checks be performed to ensure that the model describes the experimental data and that the model parameters can be estimated with precision. In addition to methodologic aspects, several compartment models of apoprotein and lipid metabolism are reviewed.     

Modified minimal model for effect of physical exercise on insulin sensitivity and glucose effectiveness in type 2 diabetes and healthy human

The Bergman’s minimal model of glucose and insulin plasma levels is commonly used to analyse the results of glucose tolerance tests in humans. In this paper, we present the modified minimal model with plasma insulin compartment under the assumption that if the plasma glucose compartment drops below the basal glucose levels, the rate of insulin entering the plasma glucose compartment is zero. Insulin is cleared from the plasma insulin compartment at a rate proportional to the amount of insulin in the plasma insulin compartment. The modified minimal model was used to study the effect of physical exercise via parameters of a mathematical model to qualitative the magnitude of changes in insulin sensitivity (S _I) and glucose effectiveness (S _G) in response to exercise in type 2 diabetes and healthy human. The short-term effects of physical exercise in type 2 diabetes did not improve S _G, but markedly improved the low S _I values found in type 2 diabetes, indicating that the effects of exercise on S _I are quantitatively important in the interpretation of training-related S _I changes and may even be therapeutically useful in type 2 diabetes patients. Physical exercise is indicated either to prevent or delay the onset of type 2 diabetes or to assure a good control of type 2 diabetes by increasing insulin sensitivity.     

Solutions for the kinetic analysis of 45calcium uptake curves

The classic solutions based on specific activity curves for the kinetic analysis of ⁴⁵Ca movements in three compartment cellular systems cannot be used when the extracellular compartment is one to two orders of magnitude larger than the cellular or tissue compartments. However if the relative radioactivity curve (tracer uptake curve) is analyzed it is possible to calculate all the relevant kinetic parameters. This paper offers the solutions based on relative radioactivity measurements for the calculation of exchange rates, rate constants and compartment sizes of three compartment systems, for series and parallel cases, for closed and open systems.     

Uptake of horseradish peroxydase by the testis of Locusta migratoria during the last larval instar; relation with variations of ecdysteroid levels in haemolymph

Summary By using horseradish peroxydase (HRP) as a tracer, it is shown that the gonial region of the locust testis is an open compartment which is almost always freely penetrated by the tracer. During the last larval instar, however, the penetration of HRP decreases and ceases at the time when high levels of ecdysteroids are detected in the haemolymph by radioimmunoassay. A cause and effect relationship between tracer uptake and hormonal level could not be demonstrated by the experiments carried out up to now. From ultrastructural observations of the testis, it is concluded that the temporary isolation of the gonial compartment is not based upon any morphological structure which could act as a barrier. Penetration of the macromolecule is considered as the expression of an active uptake by the testis and the short period of nonpenetrability as a state of inertia whose significance remains to be discovered.     

Compartmental models with restricted movement

A restriction is imposed on the number of particles that can possibly move at any time from a compartment, so that any other particles present in the compartment must wait until such particles have moved out. The equations for such a system are formulated and the solution is given for a single compartment system; increased variability of the compartmental particle count is one effect of this restriction.     

A comparison of methods for determining compartmental analysis parameters

The traditional method for determining compartmental analysis parameters relies on a visual selection of data points to be used for regression of data from each cellular compartment. This method is appropriate when the compartments are kinetically discrete and are easily discernible. However, where treatment effects on compartment parameters are being evaluated, a more objective method for determining initial parameters is desirable.

Three methods were examined for determining initial isotopic contents and half-times of ⁸⁶Rb elution from cellular compartments using theoretical data with known parameters. Experimental data from roots of Douglas fir (Pseudotsuga menziesii [Mirb.] Franco) and barley (Hordeum vulgare L.) intact seedlings were also used. The three methods were a visually assisted, linear regression on data of semilog plot of isotope elution versus time, a microcomputer-assisted, linear regression on semilog plot where maximization of the square of the correlation coefficient (r²) was the criterion to determine data points needed for each regression and a mainframe computer-assisted, direct nonlinear regression on elution data using a model of the sum of three exponential decay functions. The visual method resulted in the least accurate estimates of compartmental analysis parameters. The microcomputer-assisted and nonlinear regression methods calculated the parameters equally well.