Hidden pools,hidden modes,and visible repeated eigenvalues in compartmental models

In compartmental analysis, when time-series data are initially fitted by sums of exponential functions, it is usually assumed that the eigenvalues are real and distinct and the number of pools (compartments) equals the number of exponentials. However, repeated real or complex eigenvalues, although more difficult to detect, may be inherent in the data, and the number of pools may be larger than the number of exponentials. In order to describe compartmental models with such properties, the visible multiplicity of eigenvalues and concepts of hidden pools and visible and hidden modes are defined. It is then shown that if a model is state observable, each mode is visible (not hidden) in the zero-input response for some choice of initial state, but not conversely, and also that the visible multiplicity of eigenvalues is determined by the submodel of input-output connectable compartments. Compartmental models are analyzed, using their decomposition into what we define as strongly connected components. An upper bound is given for the visible multiplicity of eigenvalues in terms of the model's strongly connected components. For models with one input and one output, this bound is shown to be attained for what we call generalized trees.     

On impulse response functions computed from dynamic contrast-enhanced image data by algebraic deconvolution and compartmental modeling

Concentration-time courses measured by dynamic contrast-enhanced (DCE) imaging can be described by a convolution of the arterial input with an impulse response function, Q_T(t), characterizing tissue microcirculation. Data analysis is based on two different approaches: computation of Q_T(t) by algebraic deconvolution (AD) and subsequent evaluation according to the indicator dilution theory (IDT) or parameterization of Q_T(t) by analytical expressions derived by compartmental modeling. Pitfalls of both strategies will be addressed in this study.Tissue data acquired by DCE-CT in patients with head-and-neck cancer and simulated by a reference model (MMID4) were analyzed by a two-compartment model (TCM), a permeability-limited two-compartment model (PL-TCM) and AD. Additionally, MMID4 was used to compute the ‘true’ response function that corresponds to the simulated tumor data.TCM and AD yielded accurate fits, whereas PL-TCM performed worse. Nevertheless, the corresponding response functions diverge markedly. The response curves obtained by TCM decrease exponentially in the early perfusion phase and overestimate the tissue perfusion, Q_T(0). AD also resulted in response curves starting with a negative slope and not – as the ‘true’ response function in accordance with the IDT – with a horizontal plateau. They are thus not valid responses in the sense of the IDT that can be used unconditionally for parameter estimation.Response functions differing considerably in shape can result in virtually identical tissue curves. This non-uniqueness makes a strong argument not to use algebraic but rather analytical deconvolution to reduce the class of solutions to representatives that are in accordance with a-priori knowledge. To avoid misinterpretations and systematic errors, users must be aware of the pitfalls inherent to the different concepts.     

Petri Net and Probabilistic Model Checking Based Approach for the Modelling,Simulation and Verification of Internet Worm Propagation

Internet worms are analogous to biological viruses since they can infect a host and have the ability to propagate through a chosen medium. To prevent the spread of a worm or to grasp how to regulate a prevailing worm, compartmental models are commonly used as a means to examine and understand the patterns and mechanisms of a worm spread. However, one of the greatest challenge is to produce methods to verify and validate the behavioural properties of a compartmental model. This is why in this study we suggest a framework based on Petri Nets and Model Checking through which we can meticulously examine and validate these models. We investigate Susceptible-Exposed-Infectious-Recovered (SEIR) model and propose a new model Susceptible-Exposed-Infectious-Recovered-Delayed-Quarantined (Susceptible/Recovered) (SEIDQR(S/I)) along with hybrid quarantine strategy, which is then constructed and analysed using Stochastic Petri Nets and Continuous Time Markov Chain. The analysis shows that the hybrid quarantine strategy is extremely effective in reducing the risk of propagating the worm. Through Model Checking, we gained insight into the functionality of compartmental models. Model Checking results validate simulation ones well, which fully support the proposed framework.     

The role of nonreal eigenvalues in the identification of cycles in a compartmental system

The paper concerns the relationship between the cycles in the graph of a compartmental system and the modes of the impulse response function associated with an input-output experiment. Suppose that there is at least one oscillatory mode, e^μtcos(vt - α). Let e^?t be the slowest mode. The main result is that the system contains a cycle of length 3 or longer and that the length of the longest cycle is at least π/tan^-1[¦v¦/(?-μ)]. The paper also deals with the problem of estimating the cycle length from discrete data.     

Hidden oscillations in generalized linear mammaillary compartmental models

Oscillations due to complex eigenvalues, known to exist but difficult to detect, are sometimes totally hidden in the output of compartmental models, i.e., none of their modes appear in the output. An example is constructed of a class of linear compartmental models with complex eigenvalues, which have oscillating modes appearing in the output for some single-pool-input/single-pool-output (SpISpO) configurations, while for other such configurations all oscillations are totally hidden in the output. To generate the example, generalized mammillary compartmental models are defined in which a central pool exchanges with peripheral submodels called clusters, through individual connector pools, and their transfer functions are calculated corresponding to all SpISpO configurations. When such a model is repetitive, i.e., when it has identical peripheral clusters, and the input or the output is in the central pool, then it is zero-state equivalent, up to a multiplicative constant, with a reduced model having one peripheral cluster only. We analyze the visibility of an eigenvalue, i.e., whether or not the modes associated with it appear in the output, for repetitive generalized mammillary models. Sufficient conditions are given for such models to have oscillating modes appearing in the impulse response for some input/output configurations, while for other such configurations all oscillations are totally hidden, i.e., none appear in the output. A particularly interesting example is presented of a class of linear models with complex eigenvalues satisfying these conditions. This class has the structure of nonlinear models used to describe the process of protein synthesis and turnover.     

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.     

Non-invasive tracking of peripheral ventilatory response to carbon dioxide

We propose a new method for quantifying the ventilatory sensitivity of the peripheral chemoreceptors to changes in CO₂ through analysis of the natural breath-to-breath variations in ventilation (y) and alveolar Pco₂ (x). This technique is truly non-invasive in that the need for administering CO₂-enriched mixtures is circumvented, and peripheral chemosensitivity is assessed while the respiratory control system operates in its normal eucapnic state, unperturbed by external interventions. The method is based on solution of the inverse problem relating the cross-correlation between alveolar Pco₂ and ventilation changes, and the autocorrelation of changes in alveolar Pco₂. Tests are performed using simulated data generated by a closed-loop respiratory control model. The impulse response of the controller and the convective delay between lungs and chemoreceptors are estimated from the data. Subsequently, the best exponential fit to the estimated impulse response yields values for the effective controller gain (G) and the associated time constant of the response. The estimates of G contain a small contribution from the control chemoreflex gain; however, the relation between changes in G and changes in peripheral gain remains unaltered. The effects of other details in the computation procedure, such as length of data sequence, maximum number of correlation lags and starting lag number, are also investigated.     

Comparison of marker models for the analysis of the volume variation and thoracoabdominal motion pattern in untrained and trained participants

Respiratory assessment and the biomechanical analysis of chest and abdomen motion during breathing can be carried out using motion capture systems. An advantage of this methodology is that it allows analysis of compartmental breathing volumes, thoraco-abdominal patterns, percentage contribution of each compartment and the coordination between compartments. In the literature, mainly, two marker models are reported, a full marker model of 89 markers placed on the trunk and a reduced marker model with 32 markers. However, in practice, positioning and post-process a large number of markers on the trunk can be time-consuming. In this study, the full marker model was compared against the one that uses a reduced number of markers, in order to evaluate (i) their capability to obtain respiratory parameters (breath-by-breath tidal volumes) and thoracoabdominal motion pattern (compartmental percentage contributions, and coordination between compartments) during quiet breathing, and (ii) their response in different groups such as trained and untrained, male and female.Although tests revealed strong correlations of the tidal volume values in all the groups (R² > 0.93), the reduced model underestimated the trunk volume compared with the 89 marker model. The highest underestimation was found in trained males (bias of 0.43 L). The three-way ANOVA test showed that the model did not influence the evaluation of compartmental contributions and the 32 marker model was adequate to distinguish thoracoabdominal breathing pattern in the studied groups.Our findings showed that the reduced marker model could be used to analyse the thoracoabdominal motion in both trained and untrained populations but performs poorly in estimating tidal volume.     

Mean residence time in multicompartmental models with time delays

Calculation of the mean residence time (MRT) of a drug in a stationary compartmental model is classically carried out from several expressions. Nevertheless, one or more time delays between compartments modify the mean residence times. It is the aim of this paper to propose a general method for MRT calculations, in any n-compartmental models which may include time delays. As examples, catenary and mammillary models are considered.     

Inositol 1,4,5- Trisphosphate Receptor Function in Drosophila Insulin Producing Cells

The Inositol 1,4,5- trisphosphate receptor (InsP₃R) is an intracellular ligand gated channel that releases calcium from intracellular stores in response to extracellular signals. To identify and understand physiological processes and behavior that depends on the InsP₃ signaling pathway at a systemic level, we are studying Drosophila mutants for the InsP₃R (itpr) gene. Here, we show that growth defects precede larval lethality and both are a consequence of the inability to feed normally. Moreover, restoring InsP₃R function in insulin producing cells (IPCs) in the larval brain rescues the feeding deficit, growth and lethality in the itpr mutants to a significant extent. We have previously demonstrated a critical requirement for InsP₃R activity in neuronal cells, specifically in aminergic interneurons, for larval viability. Processes from the IPCs and aminergic domain are closely apposed in the third instar larval brain with no visible cellular overlap. Ubiquitous depletion of itpr by dsRNA results in feeding deficits leading to larval lethality similar to the itpr mutant phenotype. However, when itpr is depleted specifically in IPCs or aminergic neurons, the larvae are viable. These data support a model where InsP₃R activity in non-overlapping neuronal domains independently rescues larval itpr phenotypes by non-cell autonomous mechanisms.     

Inter-joint coordination of kinematics and kinetics before and after total hip arthroplasty compared to asymptomatic subjects

While differences in joint kinematics and kinetics between control subjects and patients before and after total hip arthroplasty (THA) has often been studied, inter-joint coordination has not been fully characterized. We hypothesized that in patients undergoing THA, inter-joint coordination (i) is different from control subjects before surgery, (ii) changes from pre-operative to post-operative, and (iii) remains different from control subjects after surgery. Seventy-eight subjects underwent gait analysis before and ∼1 year after primary unilateral THA. 109 control subjects were age, sex, and BMI matched to the THA group. We selected a representative trial at each subjects’ self-selected walking speed from a motion analysis data repository. To assess kinematic coordination, we constructed sagittal plane hip-knee angle cyclograms, and calculated total, stance, and swing phase plot area (deg²). To assess kinetic coordination, we calculated the support moment (M_S, %wt 1 ht), the time-integral of support moment (M_S impulse, %wt 1 ht 1 t), and the relative contribution of each joint to M_S impulse (%_Hip, %_Knee, %_Ankle). We used t-tests to compare groups. Total and swing-phase cyclogram area was smaller preoperatively, but improved to control values after THA. Swing-phase area was smaller than control values after THA. M_S impulse was larger in THA subjects than controls both before and after surgery. While, the relative contribution of the hip to M_S impulse was not different from control values, the contributions of the knee and ankle were smaller. Inter-joint coordination, as measured by hip-knee angle cyclograms and M_S impulse, may be used to distinguish differences in gait mechanics between osteoarthritis and THA. Future work focusing on coordination among joints may be needed to fully restore gait function.     

Using Multi-Compartment Ensemble Modeling As an Investigative Tool of Spatially Distributed Biophysical Balances: Application to Hippocampal Oriens-Lacunosum/Moleculare (O-LM) Cells

Multi-compartmental models of neurons provide insight into the complex, integrative properties of dendrites. Because it is not feasible to experimentally determine the exact density and kinetics of each channel type in every neuronal compartment, an essential goal in developing models is to help characterize these properties. To address biological variability inherent in a given neuronal type, there has been a shift away from using hand-tuned models towards using ensembles or populations of models. In collectively capturing a neuron''s output, ensemble modeling approaches uncover important conductance balances that control neuronal dynamics. However, conductances are never entirely known for a given neuron class in terms of its types, densities, kinetics and distributions. Thus, any multi-compartment model will always be incomplete. In this work, our main goal is to use ensemble modeling as an investigative tool of a neuron''s biophysical balances, where the cycling between experiment and model is a design criterion from the start. We consider oriens-lacunosum/moleculare (O-LM) interneurons, a prominent interneuron subtype that plays an essential gating role of information flow in hippocampus. O-LM cells express the hyperpolarization-activated current (I _h). Although dendritic I _h could have a major influence on the integrative properties of O-LM cells, the compartmental distribution of I _h on O-LM dendrites is not known. Using a high-performance computing cluster, we generated a database of models that included those with or without dendritic I _h. A range of conductance values for nine different conductance types were used, and different morphologies explored. Models were quantified and ranked based on minimal error compared to a dataset of O-LM cell electrophysiological properties. Co-regulatory balances between conductances were revealed, two of which were dependent on the presence of dendritic I _h. These findings inform future experiments that differentiate between somatic and dendritic I _h, thereby continuing a cycle between model and experiment.     

A Matrix model for Predicting Foliage Weight of Trees by Age Classes

A Leslie-type matrix has been developed to model the annual growth of foliage on individual trees for any species that retains its foliage for more than one year. In order to apply the model to any given species, one must know: (1) age-specific foliage photosynthetic efficiencies; (2) the proportion of foliage of each age class that remains to become one year older; and (3) the diameter of the tree at any two points in time, several years apart. These diameters are used to estimate the maximum eigenvalue of the matrix, which is needed in the determination of the specific numerical matrix which corresponds to a particular individual tree. Although the model is a general one, applicable to any species that has age classes of foliage, it is developed with occasional references to balsam fir, Abies balsamea (L.) Mill. For a data set of 66 dominant and codominant balsam fir trees, the model predicted the weights of the first three age classes of foliage accurately (less than 10% error) over a 6-year period. There was a 23% error for all age classes combined.     

A spatio-temporal model to describe the spread of Salmonella within a laying flock

Salmonella is one of the major sources of toxi-infection in humans, most often because of consumption of poultry products. The main reason for this association is the presence in hen flocks of silent carriers, i.e. animals harboring Salmonella without expressing any visible symptoms. Many prophylactic means have been developed to reduce the prevalence of Salmonella carrier-state. While none allows a total reduction of the risk, synergy could result in a drastic reduction of it. Evaluating the risk by modeling would be very useful to estimate such gain in food safety. Here, we propose an individual-based model which describes the spatio-temporal spread of Salmonella within a laying flock and takes into account the host response to bacterial infection. The model includes the individual bacterial load and the animals’ ability to reduce it thanks to the immune response, i.e. maximum bacterial dose that the animals may resist without long term carriage and, when carriers, length of bacterial clearance. For model validation, we simulated the Salmonella spread under published experimental conditions. There was a good agreement between simulated and observed published data. This model will thus allow studying the effects, on the spatiotemporal distribution of the bacteria, of both mean and variability of different elements of host response.     

Modelling spatial landscape complexity using the Levenshtein algorithm

A new measure (C_L) of spatial/structural landscape complexity is developed in this paper, based on the Levenshtein algorithm used in Computer Science and Bioinformatics for string comparisons. The Levenshtein distance (or edit distance) between two strings of symbols is the minimum of all possible replacements, deletions and insertions necessary to convert one string into the other. In this paper, it is shown how this measure can be applicable on raster landscape maps of any size or shape. Calculations and applications are shown on model and real landscapes. The main advantages of this measure for structural (spatial) landscape analysis are the following: it is easily applicable; it can be compared to its maximum value (depending on the grid resolution); it can be used to compare structural/spatial complexities between landscapes; it is applicable to raster landscape maps of any shape; and it can be used to calculate changes in landscape complexity over time. At the level of ecological practice, it may aid in landscape monitoring, management and planning, by identifying areas of higher structural landscape complexity, which may deserve greater attention in the process of landscape conservation.     

Chloride and Salicylate Influence Prestin-dependent Specific Membrane Capacitance: SUPPORT FOR THE AREA MOTOR MODEL*

The outer hair cell is electromotile, its membrane motor identified as the protein SLC26a5 (prestin). An area motor model, based on two-state Boltzmann statistics, was developed about two decades ago and derives from the observation that outer hair cell surface area is voltage-dependent. Indeed, aside from the nonlinear capacitance imparted by the voltage sensor charge movement of prestin, linear capacitance (C_lin) also displays voltage dependence as motors move between expanded and compact states. Naturally, motor surface area changes alter membrane capacitance. Unit linear motor capacitance fluctuation (δC_sa) is on the order of 140 zeptofarads. A recent three-state model of prestin provides an alternative view, suggesting that voltage-dependent linear capacitance changes are not real but only apparent because the two component Boltzmann functions shift their midpoint voltages (V_h) in opposite directions during treatment with salicylate, a known competitor of required chloride binding. We show here using manipulations of nonlinear capacitance with both salicylate and chloride that an enhanced area motor model, including augmented δC_sa by salicylate, can accurately account for our novel findings. We also show that although the three-state model implicitly avoids measuring voltage-dependent motor capacitance, it registers δC_sa effects as a byproduct of its assessment of C_lin, which increases during salicylate treatment as motors are locked in the expanded state. The area motor model, in contrast, captures the characteristics of the voltage dependence of δC_sa, leading to a better understanding of prestin.     

Estimating the Basic Reproductive Number (R0) for African Swine Fever Virus (ASFV) Transmission between Pig Herds in Uganda

African swine fever (ASF) is a highly contagious, lethal and economically devastating haemorrhagic disease of domestic pigs. Insights into the dynamics and scale of virus transmission can be obtained from estimates of the basic reproduction number (R₀). We estimate R₀ for ASF virus in small holder, free-range pig production system in Gulu, Uganda. The estimation was based on data collected from outbreaks that affected 43 villages (out of the 289 villages with an overall pig population of 26,570) between April 2010 and November 2011. A total of 211 outbreaks met the criteria for inclusion in the study. Three methods were used, specifically; (i) GIS- based identification of the nearest infectious neighbour based on the Euclidean distance between outbreaks, (ii) epidemic doubling time, and (iii) a compartmental susceptible-infectious (SI) model. For implementation of the SI model, three approaches were used namely; curve fitting (CF), a linear regression model (LRM) and the SI/N proportion. The R₀ estimates from the nearest infectious neighbour and epidemic doubling time methods were 3.24 and 1.63 respectively. Estimates from the SI-based method were 1.58 for the CF approach, 1.90 for the LRM, and 1.77 for the SI/N proportion. Since all these values were above one, they predict the observed persistence of the virus in the population. We hypothesize that the observed variation in the estimates is a consequence of the data used. Higher resolution and temporally better defined data would likely reduce this variation. This is the first estimate of R₀ for ASFV in a free range smallholder pig keeping system in sub-Saharan Africa and highlights the requirement for more efficient application of available disease control measures.     

Putative involvement of Thioredoxin h in early response to gravitropic stimulation of poplar stems

Gravity perception and gravitropic response are essential for plant development. In herbaceous species, it is widely accepted that one of the primary events in gravity perception involves the displacement of amyloplasts within specialized cells. However, the early signaling events leading to stem reorientation are not fully known, especially in woody species in which primary and secondary growth occur. Thirty-six percent of the identified proteins that were differentially expressed after gravistimulation were established as potential Thioredoxin targets. In addition, Thioredoxin h expression was induced following gravistimulation. In situ immunolocalization indicated that Thioredoxin h protein co-localized with the amyloplasts located in the endodermal cells. These investigations suggest the involvement of Thioredoxin h in the first events of signal transduction in inclined poplar stems, leading to reaction wood formation.     

Design and analysis of experiments on mutagenicity. III. Assays for X-chromosomal aneurploidy induced in Drosophila oocytes.

One of the simpler methods available for detecting the induction of aneuploidy in Drosophila involves the exposure to a suspected mutagen of females homozygous for a readily visible sex-linked recessive mutant allele. The treated females are mated to wild-type males, and the F₁ flies are scored for exceptional progeny (mutant ♀♀ and wild-type ♂♂). The exceptional progeny result from nondisjunction and/or chromosome loss of the X-chromosomes during oogenesis. A mathematical model is presented that describes the “fate” of primary oocytes and which allows one to derive separate estimates of the rates of nondisjunction and chromosome loss during oogenesis. Chromosome loss in this model is defined as the production of nullo-X eggs by any means other than nondisjunction. The model allows for differential viabilities among F₁ genotypes and also allows for the numbers of functional X-bearing and Y-bearing sperm from the male parents to differ from a 1:1 ratio. Statistical procedures are presented that enable one to compare experimental and control groups for rates of nondisjunction and chromosome loss. Interestingly, the spontaneous rate of nondisjunction of X-chromosomes during oogenesis is found to be several times that of chromosome loss.