Parameter estimation in a stochastic model of the tubuloglomerular feedback mechanism in a rat nephron

A key parameter in the understanding of renal hemodynamics is the gain of the feedback function in the tubuloglomerular feedback mechanism. A dynamic model of autoregulation of renal blood flow and glomerular filtration rate has been extended to include a stochastic differential equations model of one of the main parameters that determines feedback gain. The model reproduces fluctuations and irregularities in the tubular pressure oscillations that the former deterministic models failed to describe. This approach assumes that the gain exhibits spontaneous erratic variations that can be explained by a variety of influences, which change over time (blood pressure, hormone levels, etc.). To estimate the key parameters of the model we have developed a new estimation method based on the oscillatory behavior of the data. The dynamics is characterized by the spectral density, which has been estimated for the observed time series, and numerically approximated for the model. The parameters have then been estimated by the least squares distance between data and model spectral densities. To evaluate the estimation procedure measurements of the proximal tubular pressure from 35 nephrons in 16 rat kidneys have been analyzed, and the parameters characterizing the gain and the delay have been estimated. There was good agreement between the estimated values, and the values obtained for the same parameters in independent, previously published experiments.     

Modeling the euglycemic hyperinsulinemic clamp by stochastic differential equations

The Euglycemic Hyperinsulinemic Clamp (EHC) is the most widely used experimental procedure for the determination of insulin sensitivity. In the present study, 16 subjects with BMI between 18.5 and 63.6 kg/m² have been studied with a long-duration (5 hours) EHC. In order to explain the oscillations of glycemia occurring in response to the hyperinsulinization and to the continuous glucose infusion at varying speeds, we first hypothesized a system of ordinary differential equations (ODEs), with limited success. We then extended the model and represented the experiment using a system of stochastic differential equations (SDEs). The latter allow for distinction between (i) random variation imputable to observation error and (ii) system noise (intrinsic variability of the metabolic system), due to a variety of influences which change over time. The stochastic model of the EHC was fitted to data and the system noise was estimated by means of a (simulated) maximum likelihood procedure, for a series of different hypothetical measurement error values. We showed that, for the whole range of reasonable measurement error values: (i) the system noise estimates are non-negligible; and (ii) these estimates are robust to changes in the likely value of the measurement error. Explicit expression of system noise is physiologically relevant in this case, since glucose uptake rate is known to be affected by a host of additive influences, usually neglected when modeling metabolism. While in some of the studied subjects system noise appeared to only marginally affect the dynamics, in others the system appeared to be driven more by the erratic oscillations in tissue glucose transport rather than by the overall glucose-insulin control system. It is possible that the quantitative relevance of the unexpressed effects (system noise) should be considered in other physiological situations, represented so far only with deterministic models.The work was supported by grants from the Danish Medical Research Council and the Lundbeck Foundation to S. Ditlevsen.     

Process noise: an explanation for the fluctuations in the immune response during acute viral infection

The parameters of the immune response dynamics are usually estimated by the use of deterministic ordinary differential equations that relate data trends to parameter values. Since the physical basis of the response is stochastic, we are investigating the intensity of the data fluctuations resulting from the intrinsic response stochasticity, the so-called process noise. Dealing with the CD8+ T-cell responses of virus-infected mice, we find that the process noise influence cannot be neglected and we propose a parameter estimation approach that includes the process noise stochastic fluctuations. We show that the variations in data can be explained completely by the process noise. This explanation is an alternative to the one resulting from standard modeling approaches which say that the difference among individual immune responses is the consequence of the difference in parameter values.     

Metabolism of dicarboxylic acids in rat hepatocytes

[carboxyl-14C]Dodecanedioic acid (DC12) is metabolized in hepatocytes at a rate about two thirds that of [1-14C]palmitate. Shorter dicarboxylates (sebacic (DC10), suberic (DC8), and adipic (DC6) acid) are formed, mainly DC6, less DC8 and only a little DC10. In hepatocytes from clofibrate-treated rats, more polar products account for most of the breakdown products, presumably because the beta-oxidation proceeds all the way to succinate and acetyl-CoA. [carboxyl-14C]Suberic acid (DC8) is oxidized at a rate only one fifth that of dodecanedioic acid. (+)-Decanoylcarnitine inhibits palmitate oxidation but not the oxidation of dodecanedioic acid. At low concentrations of [carboxyl-14C]dodecanedioic acid or of [1-14C]palmitate, acetylsulfanilamide is more efficiently labeled by the former. High concentrations of dodecanedioic acid inhibit palmitate oxidation and the acetylation of sulfanilamide, presumably because their CoA-esters accumulate in the cytosol. These results indicate that medium-chain dicarboxylic acids are beta-oxidized mainly in the peroxisomes.     

Parameter estimation for a mathematical model of the cell cycle in frog eggs.

Parameter values for a kinetic model of the nuclear replication-division cycle in frog eggs are estimated by fitting solutions of the kinetic equations (nonlinear ordinary differential equations) to a suite of experimental observations. A set of optimal parameter values is found by minimizing an objective function defined as the orthogonal distance between the data and the model. The differential equations are solved by LSODAR and the objective function is minimized by ODRPACK. The optimal parameter values are close to the "guesstimates" of the modelers who first studied this problem. These tools are sufficiently general to attack more complicated problems, where guesstimation is impractical or unreliable.     

Approximate linear confidence and curvature of a kinetic model of dodecanedioic acid in humans

Dicarboxylic acids with an even number of carbon atoms have been proposed as an alternate energy substrate for enteral or parenteral nutrition in the acutely ill patient, due to their water solubility and their yielding TCA cycle intermediates upon beta-oxidation. In the present work, a nonlinear compartmental model of the kinetics of dodecanedioic acid is developed, and its parameters are estimated from time concentration experimental observations obtained from six healthy volunteers undergoing a per os administration of 3 g of the substance. Although the model is linear in the transfer of the free substance from plasma to the tissues, the exchange between gut and plasma compartments is represented as a saturable function. Albumin binding is then incorporated to obtain the final model in terms of the measured total concentrations. Estimates of the model's structural parameters were computed for each experimental subject, and the usual single-subject approximate confidence regions for the parameters were derived by inversion of the Hessian at the optimum. To verify the applicability of this approximation, the nonlinearity of the expectation surface at the optimum was measured by computing the normal (intrinsic) component of curvature. Because the model curvature was excessive in all subjects, the usual approximation could not be trusted to provide acceptable approximations to the parameter confidence regions. A suitable Monte Carlo simulation yielded empirical joint parameter distributions from which the approximate parameter variances could finally be obtained.     

Diffusion models for chemotaxis: a statistical analysis of noninteractive unicellular movement.

A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a one-dimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell physiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a three-dimensional experimental setup of spherical symmetry.     

Phenomenological model of changes in evoked potentials of the sensomotor area of the large cerebral hemispheres during processing of conditioned reflexes]

A phemenological model of variations in evoked potentials during conditioning is proposed. The model consists of four linear differentional equations. Variations in a wide range of only one parameter of the system of differential equations correspond to all variations in evoked potentials. A variation of this parameter in the presence of constant disturbances at the system input gives rise to the appearance of a constant or increasing signal at the output of the system, which is characteristic for the expectancy wave.     

A PDMP model of the epithelial cell turn-over in the intestinal crypt including microbiota-derived regulations

Mechanistic models are a powerful tool to gain insights into biological processes. The parameters of such models, e.g. kinetic rate constants, usually cannot be measured directly but need to be inferred from experimental data. In this article, we study dynamical models of the translation kinetics after mRNA transfection and analyze their parameter identifiability. That is, whether parameters can be uniquely determined from perfect or realistic data in theory and practice. Previous studies have considered ordinary differential equation (ODE) models of the process, and here we formulate a stochastic differential equation (SDE) model. For both model types, we consider structural identifiability based on the model equations and practical identifiability based on simulated as well as experimental data and find that the SDE model provides better parameter identifiability than the ODE model. Moreover, our analysis shows that even for those parameters of the ODE model that are considered to be identifiable, the obtained estimates are sometimes unreliable. Overall, our study clearly demonstrates the relevance of considering different modeling approaches and that stochastic models can provide more reliable and informative results.

     

Oxidation of D-[U-(14)C] glucose and [1,12-(14)C] dodecanedioic acid by pancreatic islets from Goto-Kakizaki rats.

In pancreatic islets from hereditarily diabetic GK rats, [1,12 -(14)C] dodecanedioic acid (5.0 mM) was oxidized at a rate representing about 5 % of that of D-[U - (14)C] glucose (8.3 mM). Dioic acid and hexose failed to exert any significant reciprocal effects on their respective oxidation. The production of (14)CO(2) from [1,12 -(14)C] dodecanedioic acid was proportional to its concentration in the 0.2 - 5.0 mM range. These results were essentially comparable to those obtained in islets from control rats. They extend, therefore, to GK rats the knowledge that dodecanedioic acid acts as a nutrient in pancreatic islet cells.     

Studying and approximating spatio-temporal models for epidemic spread and control

A class of simple spatio-temporal stochastic models for the spread and control of plant disease is investigated. We consider a lattice-based susceptible-infected model in which the infection of a host occurs through two distinct processes: a background infective challenge representing primary infection from external sources, and a short-range interaction representing the secondary infection of susceptibles by infectives within the population. Recent data-modelling studies have suggested that the above model may describe the spread of aphid-borne virus diseases in orchards. In addition, we extend the model to represent the effects of different control strategies involving replantation (or recovery). The Contact Process is a particular case of this model. The behaviour of the model has been studied using Cellular-Automata simulations. An alternative approach is to formulate a set of deterministic differential equations that captures the essential dynamics of the stochastic system. Approximate solutions to this set of equations, describing the time evolution over the whole parameter range, have been obtained using the pairwise approximation (PA) as well as the most commonly used mean-field approximation (MF). Comparison with simulation results shows that PA is significantly superior to MF, predicting accurately both transient and long-run, stationary behaviour over relevant parts of the parameter space. The conditions for the validity of the approximations to the present model and extensions thereof are discussed.     

A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability

A discrete-time Markov chain model, a continuous-time Markov chain model, and a stochastic differential equation model are compared for a population experiencing demographic and environmental variability. It is assumed that the environment produces random changes in the per capita birth and death rates, which are independent from the inherent random (demographic) variations in the number of births and deaths for any time interval. An existence and uniqueness result is proved for the stochastic differential equation system. Similarities between the models are demonstrated analytically and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models satisfy certain consistency conditions.     

Modelling malic acid accumulation in fruits: relationships with organic acids, potassium, and temperature

Malic acid production, degradation, and storage during fruit development have been modelled. The model assumes that malic acid content is determined essentially by the conditions of its storage in the mesocarp cells, and provides a simplified representation of the mechanisms involved in the accumulation of malate in the vacuole and their regulation by thermodynamic constraints. Solving the corresponding system of equations made it possible to predict the malic acid content of the fruit as a function of organic acids, potassium concentration, and temperature. The model was applied to peach fruit, and parameters were estimated from the data of fruit development monitored over 2 years. The predictions were in good agreement with experimental data. Simulations were performed to analyse the behaviour of the model in response to variations in composition and temperature.     

Metabolic conversion of dicarboxylic acids to succinate in rat liver homogenates. A stable isotope tracer study

The metabolic conversion of dicarboxylic acids into succinate and other gluconeogenic intermediates in rat liver homogenates was investigated using [1,2,4-13C4]dodecanedioic acid as tracer. Isotope enrichments in 3-hydroxybutyrate, succinate, fumarate, and malate, as well as dicarboxylates (dodecanedioic, sebacic, suberic, and adipic acids) were measured with selected ion monitoring capillary column gas chromatograph-mass spectrometry. Significant enrichment in the M + 4 (four labeled carbons) ion of succinate (0.4-2.9%) was detected, unequivocally demonstrating the direct conversion of dicarboxylate into succinate. In addition, significant enrichment of the M + 2 ion of succinate was also observed. This labeled species was generated from labeled acetyl-CoA through the tricarboxylic acid cycle. The partition of acetyl-CoA into the tricarboxylic acid cycle relative to ketone body formation was higher in the beta oxidation of dicarboxylate than monocarboxylate. Therefore, in addition to the production of succinate, the beta oxidation of dodecanedioate resulted in the channeling of the acetyl-CoA produced to the tricarboxylic acid cycle instead of to acetoacetate production. The enrichments in lower chain dicarboxylates are consistent with a partial bidirectional beta oxidation of dodecanedioic acid. In addition to the expected M + 0 and M + 4 labels, significant M + 2 species were detected in suberic and adipic acids. These M + 2-labeled species were produced from the released free dicarboxylate intermediates which were then reactivated and metabolized. In these experiments, the overall succinate production was derived 4% from the direct conversion of dodecanedioic acid and 11% from the indirect route via acetyl-CoA through tricarboxylic acid.     

Weak convergence of a sequence of stochastic difference equations to a stochastic ordinary differential equation

We consider a sequence of discrete parameter stochastic processes defined by solutions to stochastic difference equations. A condition is given that this sequence converges weakly to a continuous parameter process defined by solutions to a stochastic ordinary differential equation. Applying this result, two limit theorems related to population biology are proved. Random parameters in stochastic difference equations are autocorrelated stationary Gaussian processes in the first case. They are jump-type Markov processes in the second case. We discuss a problem of continuous time approximations for discrete time models in random environments.     

Parameter estimation techniques for interaction and redistribution models: a predator-prey example

Summary The use of parameter estimation techniques for partial differential equations is illustrated using a predatorprey model. Whereas ecologists have often estimated parameters in models, they have not previously been able to do so for models that describe interactions in heterogeneous environments. The techniques we describe for partial differential equations will be generally useful for models of interacting species in spatially complex environments and for models that include the movement of organisms. We demonstrate our methods using field data from a ladybird beetle (Coccinella septempunctata) and aphid (Uroleucon nigrotuberculatum) interaction. Our parameter estimation algorithms can be employed to identify models that explain better than 80% of the observed variance in aphid and ladybird densities. Such parameter estimation techniques can bridge the gap between detail-rich experimental studies and abstract mathematical models. By relating the particular bestfit models identified from our experimental data to other information on Coccinella behavior, we conclude that a term describing local taxis of ladybirds towards prey (aphids in this case) is needed in the model.     

Spike trains in a stochastic Hodgkin-Huxley system

We consider a standard Hodgkin-Huxley model neuron with a Gaussian white noise input current with drift parameter mu and variance parameter sigma(2). Partial differential equations of second order are obtained for the first two moments of the time taken to spike from (any) initial state, as functions of the initial values. The analytical theory for a 2-component (V,m) approximation is also considered. Let mu(c) (approximately 4.15) be the critical value of mu for firing when noise is absent. Large sample simulation results are obtained for mumu(c), for many values of sigma between 0 and 25. For the time to spike, the 2-component approximation is accurate for all sigma when mu=10, for sigma>7 when mu=5 and only when sigma>15 when mu=2. When mumu(c), most paths show similar behavior and the moments exhibit smoothly changing behavior as sigma increases. Thus there are a different number of regimes depending on the magnitude of mu relative to mu(c): one when mu is small and when mu is large; but three when mu is close to and above mu(c). Both for the Hodgkin-Huxley (HH) system and the 2-component approximation, and regardless of the value of mu, the CV tends to about 1.3 at the largest value (25) of sigma considered. We also discuss in detail the problem of determining the interspike interval and give an accurate method for estimating this random variable by decomposing the interval into stochastic and almost deterministic components.     

Stochastic differential equation model for cerebellar granule cell excitability

Neurons in the brain express intrinsic dynamic behavior which is known to be stochastic in nature. A crucial question in building models of neuronal excitability is how to be able to mimic the dynamic behavior of the biological counterpart accurately and how to perform simulations in the fastest possible way. The well-established Hodgkin-Huxley formalism has formed to a large extent the basis for building biophysically and anatomically detailed models of neurons. However, the deterministic Hodgkin-Huxley formalism does not take into account the stochastic behavior of voltage-dependent ion channels. Ion channel stochasticity is shown to be important in adjusting the transmembrane voltage dynamics at or close to the threshold of action potential firing, at the very least in small neurons. In order to achieve a better understanding of the dynamic behavior of a neuron, a new modeling and simulation approach based on stochastic differential equations and Brownian motion is developed. The basis of the work is a deterministic one-compartmental multi-conductance model of the cerebellar granule cell. This model includes six different types of voltage-dependent conductances described by Hodgkin-Huxley formalism and simple calcium dynamics. A new model for the granule cell is developed by incorporating stochasticity inherently present in the ion channel function into the gating variables of conductances. With the new stochastic model, the irregular electrophysiological activity of an in vitro granule cell is reproduced accurately, with the same parameter values for which the membrane potential of the original deterministic model exhibits regular behavior. The irregular electrophysiological activity includes experimentally observed random subthreshold oscillations, occasional spontaneous spikes, and clusters of action potentials. As a conclusion, the new stochastic differential equation model of the cerebellar granule cell excitability is found to expand the range of dynamics in comparison to the original deterministic model. Inclusion of stochastic elements in the operation of voltage-dependent conductances should thus be emphasized more in modeling the dynamic behavior of small neurons. Furthermore, the presented approach is valuable in providing faster computation times compared to the Markov chain type of modeling approaches and more sophisticated theoretical analysis tools compared to previously presented stochastic modeling approaches.     

Parameter estimation and change-point detection from Dynamic Contrast Enhanced MRI data using stochastic differential equations

Dynamic Contrast Enhanced imaging (DCE-imaging) following a contrast agent bolus allows the extraction of information on tissue micro-vascularization. The dynamic signals obtained from DCE-imaging are modeled by pharmacokinetic compartmental models which integrate the Arterial Input Function. These models use ordinary differential equations (ODEs) to describe the exchanges between the arterial and capillary plasma and the extravascular-extracellular space. Their least squares fitting takes into account measurement noises but fails to deal with unpredictable fluctuations due to external/internal sources of variations (patients’ anxiety, time-varying parameters, measurement errors in the input function, etc.). Adding Brownian components to the ODEs leads to stochastic differential equations (SDEs). In DCE-imaging, SDEs are discretely observed with an additional measurement noise. We propose to estimate the parameters of these noisy SDEs by maximum likelihood, using the Kalman filter. In DCE-imaging, the contrast agent injected in vein arrives in plasma with an unknown time delay. The delay parameter induces a change-point in the drift of the SDE and ODE models, which is estimated also. Estimations based on the SDE and ODE pharmacokinetic models are compared to real DCE-MRI data. They show that the use of SDE provides robustness in the estimation results. A simulation study confirms these results.