The influence of numerical error on parameter estimation and uncertainty quantification for advective PDE models.

Advective partial differential equations can be used to describe many scientific processes. Two significant sources of error that can cause difficulties in inferring parameters from experimental data on these processes include (i) noise from the measurement and collection of experimental data and (ii) numerical error in approximating the forward solution to the advection equation. How this second source of error alters parameter estimation and uncertainty quantification during an inverse problem methodology is not well understood. As a step towards a better understanding of this problem, we present both analytical and computational results concerning how a least squares cost function and parameter estimator behave in the presence of numerical error in approximating solutions to the underlying advection equation. We investigate residual patterns to derive an autocorrelative statistical model that can improve parameter estimation and confidence interval computation for first order methods. Building on our results and their general nature, we provide guidelines for practitioners to determine when numerical or experimental error is the main source of error in their inference, along with suggestions of how to efficiently improve their results.

Pattern Formation in the Longevity-Related Expression of Heat Shock Protein-16.2 in Caenorhabditis elegans.

Aging in Caenorhabditis elegans is controlled, in part, by the insulin-like signaling and heat shock response pathways. Following thermal stress, expression levels of small heat shock protein-16.2 show a spatial patterning across the 20 intestinal cells that reside along the length of the worm. Here, we present a hypothesized mechanism that could lead to this patterned response and develop a mathematical model of this system to test our hypothesis. We propose that the patterned expression of heat shock protein is caused by a diffusion-driven instability within the pseudocoelom, or fluid-filled cavity, that borders the intestinal cells in C. elegans. This instability is due to the interactions between two classes of insulin-like peptides that serve antagonistic roles. We examine output from the developed model and compare it to experimental data on heat shock protein expression. Given biologically bounded parameters, the model presented is capable of producing patterns similar to what is observed experimentally and provides a first step in mathematically modeling aging-related mechanisms in C. elegans.

Determining equilibrium osmolarity in poly(ethylene glycol)/chondrotin sulfate gels mimicking articular cartilage.

We present an experimentally guided, multi-phase, multi-species polyelectrolyte gel model to make qualitative predictions on the equilibrium electro-chemical properties of articular cartilage. The mixture theory consists of two different types of polymers: poly(ethylene gylcol) (PEG), chondrotin sulfate (ChS), water (acting as solvent) and several different ions: H(+), Na(+), Cl(-). The polymer chains have covalent cross-links whose effect on the swelling kinetics is modeled via Doi rubber elasticity theory. Numerical studies on equilibrium polymer volume fraction and net osmolarity (difference in the solute concentration across the gel) show a complex interplay between ionic bath concentrations, pH, cross-link fraction and the average charge per monomer. Generally speaking, swelling is aided due to a higher average charge per monomer (or a higher particle fraction of ChS, the charged component of the polymer), low solute concentration in the bath, a high pH or a low cross-link fraction. A peculiar case arises at higher values of cross-link fraction, where it is observed that increasing the average charge per monomer leads to gel deswelling.

Determination of personalized diabetes treatment plans using a two-delay model.

Diabetes cases worldwide have risen steadily over the past few decades, lending urgency to the search for more efficient, effective, and personalized ways to treat the disease. Current treatment strategies, however, may fail to maintain oscillations in blood glucose concentration that naturally occur multiple times per day, an important element of normal human physiology. Building upon recent successes in mathematical modeling of the human glucose-insulin system, we show that both food intake and insulin therapy likely demand increasingly precise control over insulin sensitivity if oscillations at a healthy average glucose concentration are to be maintained. We then model and describe personalized treatment options for patients with diabetes that maintain these oscillations. We predict that for a person with type II diabetes, both blood glucose levels can be controlled and healthy oscillations maintained when the patient gets an hour of daily exercise and is placed on a combination of Metformin and sulfonylurea drugs. We note that insulin therapy and an additional hour of exercise will reduce the patient׳s need for sulfonylureas. Results of a modeling analysis suggest that, with constant nutrition and controlled exercise, the blood glucose levels of a person with type I diabetes can be properly controlled with insulin infusion between 0.45 and 0.7µU/mlmin. Lastly, we note that all suggested strategies rely on existing clinical techniques and established treatment measures, and so could potentially be of immediate use in the design of an artificial pancreas.

The medial septum enhances reversal learning via opposing actions on ventral tegmental area and substantia nigra dopamine neurons.

Cognitive flexibility deficits are one of the most pervasive symptoms across psychiatric disorders, making continued investigation of the circuitry underlying this function a top priority. Medial septum (MS) lesions lead to perseverative, inflexible-type behavior; however, a role for this region in cognitive flexibility circuitry has never been examined. We activated the MS (DREADDs) and measured performance in a T-maze spatial reversal learning task in male Sprague-Dawley rats. Systemic activation of the MS (CNO) significantly decreased both trials to perform a reversal and entries into the previously baited arm. Intra-ventral subiculum CNO enhanced reversal learning in the same manner as systemic CNO and also significantly increased ventral tegmental area and decreased substantia nigra dopamine neuron population activity. Finally, co-injection of the D1 antagonist SCH23390 with CNO prevented the enhanced reversal learning performance seen in the previous two experiments. Taken together, these data suggest a key role for the MS in cognitive flexibility, and suggest that MS-mediated changes in midbrain dopamine neuron population activity could be one mechanism by which this occurs.

INVESTIGATION OF A STRUCTURED FISHER'S EQUATION WITH APPLICATIONS IN BIOCHEMISTRY.

Recent biological research has sought to understand how biochemical signaling pathways, such as the mitogen-activated protein kinase (MAPK) family, influence the migration of a population of cells during wound healing. Fisher's Equation has been used extensively to model experimental wound healing assays due to its simple nature and known traveling wave solutions. This partial differential equation with independent variables of time and space cannot account for the effects of biochemical activity on wound healing, however. To this end, we derive a structured Fisher's Equation with independent variables of time, space, and biochemical pathway activity level and prove the existence of a self-similar traveling wave solution to this equation. We exhibit that these methods also apply to a general structured reaction-diffusion equation and a chemotaxis equation. We also consider a more complicated model with different phenotypes based on MAPK activation and numerically investigate how various temporal patterns of biochemical activity can lead to increased and decreased rates of population migration.

Oral administration of a specific kynurenic acid synthesis (KAT II) inhibitor attenuates evoked glutamate release in rat prefrontal cortex.

Cognitive deficits represent core symptoms in schizophrenia (SZ) and predict patient outcome; however, they remain poorly treated by current antipsychotic drugs. Elevated levels of the endogenous alpha7 nicotinic receptor negative allosteric modulator and NMDA receptor antagonist, kynurenic acid (KYNA), are commonly seen in post-mortem tissue and cerebrospinal fluid of patients with SZ. When acutely or chronically elevated in rodents, KYNA produces cognitive deficits similar to those seen in the disease, making down-regulation of KYNA, via inhibition of kynurenine aminotransferase II (KAT II), a potential treatment strategy. We determined, in adult Wistar rats, if the orally available KAT II inhibitor BFF816 a) prevents KYNA elevations in prefrontal cortex (PFC) after a systemic kynurenine injection and b) reverses the kynurenine-induced attenuation of evoked prefrontal glutamate release caused by stimulation of the nucleus accumbens shell (NAcSh). Systemic injection of kynurenine (25 or 100 mg/kg, i.p.) increased KYNA levels in PFC (532% and 1104% of baseline, respectively). NMDA infusions (0.15 µg/0.5 µL) into NAcSh raised prefrontal glutamate levels more than 30-fold above baseline. The two doses of kynurenine reduced evoked glutamate release in PFC (by 43% and 94%, respectively, compared to NMDA alone). Co-administration of BFF816 (30 or 100 mg/kg, p.o.) with kynurenine (25 mg/kg, i.p.) attenuated the neosynthesis of KYNA and dose-dependently restored NMDA-stimulated glutamate release in the PFC (16% and 69%, respectively). The ability to prevent KYNA neosynthesis and to normalize evoked glutamate release in PFC justifies further development of KAT II inhibitors for the treatment of cognitive deficits in SZ.

Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models.

The dynamics of HIV-1 infection consist of three distinct phases starting with primary infection, then latency and finally AIDS or drug therapy. In this paper we model the dynamics of primary infection and the beginning of latency. We show that allowing for time delays in the model better predicts viral load data when compared to models with no time delays. We also find that our model of primary infection predicts the turnover rates for productively infected T cells and viral totals to be much longer than compared to data from patients receiving anti-viral drug therapy. Hence the dynamics of the infection can change dramatically from one stage to the next. However, we also show that with the data available the results are highly sensitive to the chosen model. We compare the results using analysis and Monte Carlo techniques for three different models and show how each predicts rather dramatic differences between the fitted parameters. We show, using a chi(2) test, that these differences between models are statistically significant and using a jackknifing method, we find the confidence intervals for the parameters. These differences in parameter estimations lead to widely varying conclusions about HIV pathogenesis. For instance, we find in our model with time delays the existence of a Hopf bifurcation that leads to sustained oscillations and that these oscillations could simulate the rapid turnover between viral strains and the appropriate CTL response necessary to control the virus, similar to that of a predator-prey type system.

Positive allosteric modulators of the α7 nicotinic acetylcholine receptor potentiate glutamate release in the prefrontal cortex of freely-moving rats.

Positive allosteric modulators (PAMs) of α7 nicotinic acetylcholine receptors (α7nAChRs) exhibit pro-cognitive effects in animal models of schizophrenia and are targets for the discovery of cognition-enhancing drugs. However, little is known about their in vivo mechanism of action because such studies have been performed in vitro. Here we test the hypothesis that PAMs' potentiation of glutamate release in prefrontal cortex depends upon the level of endogenous cholinergic activity. NMDA stimulation of the nucleus accumbens shell (0.05-0.30 µg in 0.5 µL) increased extracellular choline (0.87 ± 0.15 - 1.73 ± 0.31 µM) and glutamate (0.15 µg, 3.79 ± 0.87 µM) in medial prefrontal cortex, and the glutamate release was prevented by local infusions of MLA (6.75 µg, 0.19 ± 0.06 µM). The lower dose (1 mg/kg) of AVL3288 (type I) potentiated the glutamate release to a greater degree after the high dose of NMDA (0.30 µg; 84.7% increase vs AVL vehicle) versus the low dose of NMDA (0.05 µg; 24.2% increase), whereas glutamate release was inhibited when the high dose of NMDA was combined with the high dose of AVL3288 (64.2% decrease). In contrast, PNU120596 (type II) only potentiated glutamate release when the high dose (9 mg/kg) was combined with the low dose of NMDA (0.05 µg; 211% increase from PNU vehicle). Collectively, the results suggest a potential in vivo mechanism for the pro-cognitive effects of PAMs and provide the proof-of-concept for the continued focus on allosteric modulation of cortical α7nAChRs for cognition-enhancing drug development.

Incorporation of variability into the modeling of viral delays in HIV infection dynamics.

We consider classes of functional differential equation models which arise in attempts to describe temporal delays in HIV pathogenesis. In particular, we develop methods for incorporating arbitrary variability (i.e., general probability distributions) for these delays into systems that cannot readily be reduced to a finite number of coupled ordinary differential equations (as is done in the method of stages). We discuss modeling from first principles, introduce several classes of non-linear models (including discrete and distributed delays) and present a discussion of theoretical and computational approaches. We then use the resulting methodology to carry out simulations and perform parameter estimation calculations, fitting the models to a set of experimental data. Results obtained confirm the statistical significance of the presence of delays and the importance of including delays in validating mathematical models with experimental data. We also show that the models are quite sensitive to the mean of the distribution which describes the delay in viral production, whereas the variance of this distribution has relatively little impact.

Transient inactivation of the ventral hippocampus in neonatal rats impairs the mesolimbic regulation of prefrontal glutamate release in adulthood.

Cognitive deficits in schizophrenia (SZ) reflect maturational disruptions within a neural system that includes the ventral hippocampus (VH), nucleus accumbens (NAc), basal forebrain, and prefrontal cortex (PFC). A better understanding of these changes may reveal drug targets for more efficacious cognition enhancers. We have utilized an animal model in which the above distributed system is altered, during a sensitive period of development, by transiently inactivating the VH and its efferent projections. We determined the ability of NAc shell activation to evoke prefrontal glutamate release in adult male Wistar rats that had received saline (Sal) or tetrodotoxin (TTX) as neonates (PD7) or as adolescents (PD32). The nucleus accumbens shell (NAcSh) was activated by NMDA infusions (0.05-0.30 µg/0.5 µL). Basal and evoked glutamate levels were measured amperometrically using a glutamate-sensitive microelectrode. There were no differences in basal glutamate levels among the groups tested (overall 1.41 ± 0.26 uM). However, the dose-related stimulation of prefrontal glutamate levels seen in control rats treated with saline on PD7 (4.31 ± 0.22 µM after 0.15 µg) was markedly attenuated in rats treated with TTX on PD7 (0.45 ± 0.12 µM after 0.15 µg). This effect was age-dependent as infusions of TTX on PD32 did not alter the NMDA-induced increases in glutamate release (4.10 ± 0.37 µM after 0.15 µg). Collectively, these findings reveal that transient inactivation of VH transmission, during a sensitive period of development, leads to a functional mesolimbic-cortical disconnection that produces neurochemical and ultimately cognitive impairments resembling those seen in SZ.

Localized infusions of the partial alpha 7 nicotinic receptor agonist SSR180711 evoke rapid and transient increases in prefrontal glutamate release.

The ability of local infusions of the alpha 7 nicotinic acetycholine receptor (α7 nAChR) partial agonist SSR180711 to evoke glutamate release in prefrontal cortex was determined in awake rats using a microelectrode array. Infusions of SSR180711 produced dose-dependent increases in glutamate levels. The lower dose (1.0µg in 0.4µL) evoked a rapid rise (∼1.0s) in glutamate (1.41±0.30µM above baseline). The higher dose (5.0µg) produced a similarly rapid, yet larger increase (3.51±0.36µM above baseline). After each dose, the glutamate signal was cleared to basal levels within 7-18s. SSR180711-evoked glutamate was mediated by the α7 nAChR as co-infusion of the selective α7 nAChR antagonist α-bungarotoxin (10.0µM)+SSR1808711 (5.0µg) reduced the effect of 5.0µg alone by 87% (2.62 vs. 0.35µM). Finally, the clearance of the SSR180711 (5.0µg)-evoked glutamate was bidirectionally affected by drugs that inhibited (threo-beta-benzyl-oxy-aspartate (TßOA), 100.0µM) or facilitated (ceftriaxalone, 200mg/kg, i.p.) excitatory amino acid transporters. TßOA slowed both the clearance (s) and rate of clearance (µM/s) by 10-fold, particularly at the mid-late stages of the return to baseline. Ceftriaxone reduced the magnitude of the SSR180711-evoked increase by 65%. These results demonstrate that pharmacological stimulation of α7 nAChRs within the prefrontal cortex is sufficient to evoke rapid yet transient increases in glutamate levels. Such increases may underlie the cognition-enhancing effects of the drug in animals; further justifying studies on the use of α7 nAChR-positive modulators in treating cognition-impairing disorders in humans.

Klebsiella pneumoniae flocculation dynamics.

The bacterial pathogen Klebsiella pneumoniae is a cause of community- and hospital-acquired lung, urinary tract and blood stream infections. It is a common contaminant of indwelling catheters and it is theorized in that context that systemic infection follows shedding of aggregates off of surface-adherent biofilm colonies. In an effort to better understand bacterial proliferation in the host bloodstream, we develop a PDE model for the flocculation dynamics of Klebsiella pneumoniae in suspension. Existence and uniqueness results are provided, as well as a brief description of the numerical approximation scheme. We generate artificial data and illustrate the requirements to accurately identify proliferation, aggregation, and fragmentation of flocs in the experimental domain of interest.

Model selection and mixed-effects modeling of HIV infection dynamics.

We present an introduction to a model selection methodology and an application to mathematical models of in vivo HIV infection dynamics. We consider six previously published deterministic models and compare them with respect to their ability to represent HIV-infected patients undergoing reverse transcriptase mono-therapy. In the creation of the statistical model, a hierarchical mixed-effects modeling approach is employed to characterize the inter- and intra-individual variability in the patient population. We estimate the population parameters in a maximum likelihood function formulation, which is then used to calculate information theory based model selection criteria, providing a ranking of the abilities of the various models to represent patient data. The parameter fits generated by these models, furthermore, provide statistical support for the higher viral clearance rate c in Louie et al. [AIDS 17:1151-1156, 2003]. Among the candidate models, our results suggest which mathematical structures, e.g., linear versus nonlinear, best describe the data we are modeling and illustrate a framework for others to consider when modeling infectious diseases.

A parameter sensitivity methodology in the context of HIV delay equation models.

A sensitivity methodology for nonlinear delay systems arising in one class of cellular HIV infection models is presented. Theoretical foundations for a typical sensitivity investigation and illustrative computations are given.

Sensitivity analysis of a nonlinear lumped parameter model of HIV infection dynamics.

A formal sensitivity analysis is performed on a delay differential equation model for the viral dynamics of an in vivo HIV infection during protease inhibitor therapy. We present results of both a differential analysis as well as a principle component based analysis and provide evidence that suggests the exact times at which specific parameters have the most influence over the solution. We offer insight into the pairwise mathematical relationships between the productively infected T-cell death rate delta, the viral plasma clearance rate c, and the time delay tau between infection and viral production as they relate to the viral dynamics. The results support the claim that the presence of a nonzero delay has a major impact on the model dynamics. Lastly, we comment upon the inadequacies of an alternative principle component based analysis.