Compartmental systems and generating functions

Compartmental systems can be represented by direct graphs in which each node corresponds to a generating function and each arm to a transfer generating function. A homomorphism is established between a compartmental system and this representation, in analogy with that obtained through the use of the Laplace transformation. From the values obtained experimentally in a given compartment, through the solution of a difference equation, the generating function for the corresponding node can be calculated and the graph of the system can be built up within the degrees of freedom of the model. From the graph it is possible to calculate the transfer generating function between any two connected nodes, the mean permanence time in a given node, the mean transit time between two nodes, and their precursor-successor order.     

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.     

Compartmental neural simulations with spatial adaptivity

Since their inception, computational models have become increasingly complex and useful counterparts to laboratory experiments within the field of neuroscience. Today several software programs exist to solve the underlying mathematical system of equations, but such programs typically solve these equations in all parts of a cell (or network of cells) simultaneously, regardless of whether or not all of the cell is active. This approach can be inefficient if only part of the cell is active and many simulations must be performed. We have previously developed a numerical method that provides a framework for spatial adaptivity by making the computations local to individual branches rather than entire cells (Rempe and Chopp, SIAM Journal on Scientific Computing, 28: 2139–2161, 2006). Once the computation is reduced to the level of branches instead of cells, spatial adaptivity is straightforward: the active regions of the cell are detected and computational effort is focused there, while saving computations in other regions of the cell that are at or near rest. Here we apply the adaptive method to four realistic neuronal simulation scenarios and demonstrate its improved efficiency over non-adaptive methods. We find that the computational cost of the method scales with the amount of activity present in the simulation, rather than the physical size of the system being simulated. For certain problems spatial adaptivity reduces the computation time by up to 80%.     

A Stochastic Compartmental Model with Long Lasting Infusion

Most of the compartmental models in current use to model pharmacokinetic systems are deterministic. Stochastic formulations of pharmacokinetic compartmental models introduce stochasticity through either a probabilistic transfer mechanism or the randomization of the transfer rate constants. In this paper we consider a linear stochastic differential equation (LSDE) which represents a stochastic version of a one‐compartment linear model when input function undergoes random fluctuations. The solution of the LSDE, its mean value and covariance structure are derived. An explicit likelihood function is obtained either when the process is observed continuously over a period of time or when sampled data are available, as it is generally feasible. We discuss some asymptotic properties of the maximum likelihood estimators for the model parameters. Furthermore we develop expressions for two random variables of interest in pharmacokinetics: the area under the time‐concentration curve, M₀(T), and the plateau concentration, x_ss. Finally the estimation procedure is illustrated by an application to real data.     

Microbially influenced corrosion of galvanized steel pipes in aerobic water systems

Aims: To investigate the role of heterotrophic bacteria in the corrosion of galvanized steel in the presence of water. Methods and Results: Samples were taken from corroding galvanized steel pipes conveying water for specialist applications, and heterotrophic bacteria were isolated and cultured. The majority of bacteria were Gram‐negative aerobes and included Pseudomonas sp., Bacillus pumilus, Afipia spp. and Blastobacter denitrificans/Bradyrhizobium japonicum. Zinc tolerance was assessed through growth and zinc disc diffusion experiments. In general, zinc negatively influenced growth rates. An unidentified yeast also isolated from the system demonstrated a high tolerance to zinc at concentrations up to 4 g l^?1. Coupon experiments were performed to assess corrosion by the bacteria on galvanized steel and steel coupons. The majority of isolates as pure culture biofilms (69%) accelerated corrosion of galvanized coupons, assessed as zinc release, relative to sterile control coupons (P < 0·05). Pure culture biofilms did not increase the corrosion of steel, with four isolates demonstrating protective effects. Conclusions: Pure culture biofilms of heterotrophic bacteria isolated from a corroding galvanized pipe system were found to accelerate the corrosion of galvanized steel coupons. Significance and Impact of the Study: Microbially influenced corrosion is a potential contributor to sporadically occurring failures in galvanized steel systems containing water. Management strategies should consider microbial control as a means for corrosion prevention in these systems.     

Compartmental models with spatial diffusion: estimation for bone-seeking tracers

A compartmentlike model is developed for tracers in which bone-volume diffusion plays an important role in body distribution (e.g. bone-volume-seeking metals). The model requires solution of an infinite eigensystem. Approximations are presented for cylindrical diffusion in canalicular territory. An application to lead metabolism in beagle dogs suggests that finite truncation of the system of equations provides an adequate approximation for routine use in computer programs for compartmental-parameter estimation. The model is consistent with both power-law and exponential mixture retention functions.     

Compartmental and intracompartmental regulation of aging

Individual differences of aging occur in systems of regulation and communication. Different types of suppressor capacity can change in opposite directions as an individual ages. The progression of these changes in suppressor capacity can be changed to different extents by hormones or diets differing in fatty acid composition. It seems reasonable to conclude that these two activities depend on different precursors which age under the control of two different genes, although the two precursors are almost certainly derived from a common stem cell line. A variety of molecules and activities undergo age-related reductions, which are completed by middle age; the resulting levels of a given gene product are remarkably similar in different individuals, while the levels at a young age show very great individual differences. This correlation between the youthful quantity of a given gene product and the rate of change in later life has been designated as "economic correction." The study of multicentricity of different controlling and of functional competencies of different alleles is an important component in the development of preventive geriatric medicine.     

Compartmental models with Erlang distributed residence times and random rate coefficients

In this paper a stochastic model for a two-compartment system which incorporates Erlang residence time distributions (i.e. the residence times have the gamma distribution where the shape parameters assume integer values only) into each compartment is generalized to include random rate coefficients. Analytical forms of the model are derived for the case where the rate coefficients have gamma densities. A relationship is established between the new models and existing models that are in current practical usage.     

Compartmental models for human iodine metabolism

Compartmental models for the various aspects of human iodine metabolism are reviewed, emphasizing the role of Mones Berman in the development of this field. The review first presents published submodels for the peripheral distribution of inorganic iodine, for the thyroidal iodide trapping function, and for the peripheral distribution and metabolism of the thyroid hormones. Approaches to improving understanding of the physiology of the thyroid gland itself through compartmental modeling techniques are then discussed in more detail. The three submodels described above are incorporated into overall models of thyroid iodine metabolism after being simplified to various degrees. Previously published models for thyroid-gland radioiodine metabolism, as well as current work in progress, are illustrated by attempting to fit the models to data from a single (previously unpublished) detailed prolonged ¹²⁵I feeding experiment in a normal human subject. Published thyroid gland models reviewed include: (1) the usual presentation, where the thyroid is a single homogeneous iodine compartment; (2) the model of DeGroot and colleagues, where thyroidal iodine is presented as MIT, DIT, T3, and T4, each with an active and linked storage compartment; (3) the thyroid model developed by Berman and colleagues, with less chemical subcategorization but incorporating a delay compartment, in which a fraction of the iodinated material in the thyroid is partially or completely inaccessible to secretion during the delay; and the later updating of Berman's model to include a thyroidal iodide recirculation pool. The experimental data presented fits most of these models for the first 1–2 weeks, but the fit could not be extended to longer data collection times. To overcome this shortcoming, a new thyroid gland model is introduced. It is based on the latest Berman model but describes thyroglobulin metabolism as incorporating multiple delay compartments of various time periods. The overall fit of the long term data is better with this model construct than with any of the published models. It appears that a complex thyroidal substructure, such as that of the multidelay model under development, will be required to account for overall thyroid iodine metabolism as isotopic equilibrium in man is approached.     

Parameter Estimation for the Compartmental Model

An estimation procedure is obtained for a stochastic compartmental model. Compartmental analysis assumes that a system may be divided into homogeneous components, or compartments. The main theory for the compartmental system was studied by Matis and Hartley (1971) with a discrete population in a steady state. All the transitions among the particles are considered to be stochastic in nature. An estimation procedure, Regular Best Asymptotic Normal (RBAN), discussed by Chiang (1956) is investigated for a stochastic m-compartmental system. The detailed proof of the procedure is provided here. Asymptotic properties for the estimator has been studied and computation has been carried out on our proposed nonlinear model. The downhill simplex search method, originally developed by Nelder and Mead (1965), and applied to minimize our quadratic form is inherently nonlinear in nature, thus avoiding the need to evaluate any derivative for point estimation of the parameters. The procedure applied to an experimental situation involving two compartments gives very encouraging results.     

Compartmental architecture and dynamics of hematopoiesis

BACKGROUND: Blood cell formation is maintained by the replication of hematopoietic stem cells (HSC) that continuously feed downstream "compartments" where amplification and differentiation of cells occurs, giving rise to all blood lineages. Whereas HSC replicate slowly, committed cells replicate faster as they become more differentiated. METHODOLOGY/SIGNIFICANT FINDING: We propose a multi-compartment model of hematopoiesis, designed on the principle of cell flow conservation under stationary conditions. Cells lost from one compartment due to differentiation are replaced by cells from the upstream compartment. We assume that there is a constant relationship between cell input and output in each compartment and fix the single parameter of the model using data available for granulocyte maturation. We predict that approximately 31 mitotic events separate the HSC from the mature cells observed in the circulation. Besides estimating the number of compartments, our model allows us to estimate the size of each compartment, the rate of cell replication within each compartment, the mean time a given cell type contributes to hematopoiesis, the amplification rate in each compartment, as well as the mean time separating stem-cell replication and mature blood-cell formation. CONCLUSIONS: Despite its simplicity, the model agrees with the limited in vivo data available and can make testable predictions. In particular, our prediction of the average lifetime of a PIG-A mutated clone agrees closely with the experimental results available for the PIG-A gene mutation in healthy adults. The present elucidation of the compartment structure and dynamics of hematopoiesis may prove insightful in further understanding a variety of hematopoietic disorders.     

Compartmental study on the pharmacokinetics of D-penicillamine

Summary An eight-compartment model was developed for the pharmacokinetics of D-penicillamine. The analysis shows that a simpler model, based on the assumption of one chemical form of penicillamine only, fails and that the concept of two different forms of penicillamine must be introduced. Most probably, it concerns the disulfide of penicillamine and its mixed disulfide with cysteine. The primary distribution volume for both compounds is the extracellular fluid. Binding to plasma proteins has no essential effect on the overall kinetics. The two disulfides are handled by the kidneys in a different manner.     

Modelling Cultivar Mixtures Using SEIR Compartmental Models

A SEIR (susceptible, exposed, infectious, removed) compartmental model is constructed to represent disease progress in a two cultivar mixture. The concept of the spore pool is the means by which inoculum exchange between the constituent cultivars is represented. Infection frequencies for each cultivar are permitted to vary with that cultivar's susceptible fraction according to a power-law relationship. For each cultivar, new additions to the susceptible class balance deaths from the removed class so that the total leaf area of all four SEIR classes remains constant. It is shown that an equilibrium with non-zero diseased classes exists in a certain parameter regime. A numerical stability analysis is performed using the model equations linearised about this equilibrium. The effects of changing induced resistance parameters within the model are demonstrated graphically. It is also demonstrated that the proportion of susceptibles as a function of mixture composition has an optimum in the regime where nontrivial equilibria exist, a feature of practical interest provided that equilibrium is reached within a growing season.     

Compartmental models with multiple sources of stochastic variability: The one-compartment models with clustering

Previous compartmental models have introduced variability either at the particle or at the replicate level. This paper integrates both types of variability through the concept of clustering. The paper develops two different, general clustered models, each with time-dependent hazard rates for the clusters and for the particles within the clusters, and each with random initial number and sizes of clusters. The coefficient of variation of the total number of particles,CV[X(t)], for either model is shown to be bounded below, under very broad conditions, by the coefficient of variation of the initial number of clusters,CV[c(0)]. This high relative variability of the clustered models makes them potentially very useful in kinetic modeling. In many applications, binding and clustering are common phenomena, and two applications of the models to such phenomena are breifly outlined.     

A Compartmental Model for Activity-Dependent Dendritic Spine Branching

Dendritic spines are small, mushroom-like protrusions from the arbor of a neuron in the central nervous system. Interdependent changes in the morphology, biochemistry, and activity of spines have been associated with learning and memory. Moreover, post-mortem cortices from patients with Alzheimer’s or Parkinson’s disease exhibit biochemical and physical alterations within their dendritic arbors and a reduction in the number of dendritic spines. For over a decade, experimentalists have observed perforations in postsynaptic densities on dendritic spines after induction of long-term potentiation, a sustained enhancement of response to a brief electrical or chemical stimulus, associated with learning and memory. In more recent work, some suggest that activity-dependent intraspine calcium may regulate the surface area of the spine head, and reorganization of postsynaptic densities on the surface. In this paper, we develop a model of a dendritic spine with the ability to partition its transmission and receptor zones, as well as the entire spine head. Simulations are initially performed with fixed parameters for morphology to study electrical properties and identify parameters that increase efficacy of the synaptic connection. Equations are then introduced to incorporate calcium as a second messenger in regulating continuous changes in morphology. In the model, activity affects compartmental calcium, which regulates spine head morphology. Conversely, spine head morphology affects the level of local activity, whether the spines are modeled with passive membrane properties, or excitable membrane using Hodgkin–Huxley kinetics. Results indicate that merely separating the postsynaptic receptors on the surface of the spine may add to the diversity of circuitry, but does not change the efficacy of the synapse. However, when the surface area of the spine is a dynamic variable, efficacy of the synapse may vary continuously over time.     

GPU-Accelerated Compartmental Modeling Analysis of DCE-MRI Data from Glioblastoma Patients Treated with Bevacizumab

The compartment model analysis using medical imaging data is the well-established but extremely time consuming technique for quantifying the changes in microvascular physiology of targeted organs in clinical patients after antivascular therapies. In this paper, we present a first graphics processing unit-accelerated method for compartmental modeling of medical imaging data. Using this approach, we performed the analysis of dynamic contrast-enhanced magnetic resonance imaging data from bevacizumab-treated glioblastoma patients in less than one minute per slice without losing accuracy. This approach reduced the computation time by more than 120-fold comparing to a central processing unit-based method that performed the analogous analysis steps in serial and more than 17-fold comparing to the algorithm that optimized for central processing unit computation. The method developed in this study could be of significant utility in reducing the computational times required to assess tumor physiology from dynamic contrast-enhanced magnetic resonance imaging data in preclinical and clinical development of antivascular therapies and related fields.     

Compartmental distribution of amino acids during hemodialysis-induced hypoaminoacidemia

The intracellular concentrations of essential amino acids (EAA) in muscle are maintained relatively constant under a variety of conditions. However, the effect of a decrease in blood amino acid concentrations on intracellular concentrations is not clear. Similarly, the relation between intracellular and interstitial concentrations has not been determined in this circumstance. Thus the aim of this study was to determine the effect of hypoaminoacidemia on intracellular, interstitial, and plasma concentrations of EAA and the mechanisms responsible for the respective changes. Twelve normal pigs were investigated before and during 120 min of hemodialysis by use of stable-isotope tracer methodology, microdialysis technique, and muscle biopsies. During hemodialysis, there was a decrease in the interstitial fluid concentrations of phenylalanine, leucine, alanine, and lysine that corresponded to their decrease in plasma concentration. Nonetheless, the intracellular concentrations of these amino acids were maintained at the basal levels throughout the entire period due principally to a reduction in the rate of incorporation of amino acids into protein that was approximately equivalent to the decrease in uptake from the plasma. In conclusion, intracellular concentrations of amino acids are regulated to maintain relatively constant values, even when plasma and interstitial concentrations fall as a consequence of hemodialysis.