Post-pandemic modeling of COVID-19: Waning immunity determines recurrence frequency.

There are many factors in the current phase of the COVID-19 pandemic that signal the need for new modeling ideas. In fact, most traditional infectious disease models do not address adequately the waning immunity, in particular as new emerging variants have been able to break the immune shield acquired either by previous infection by a different strain of the virus, or by inoculation of vaccines not effective for the current variant. Furthermore, in a post-pandemic landscape in which reporting is no longer a default, it is impossible to have reliable quantitative data at the population level. Our contribution to COVID-19 post-pandemic modeling is a simple mathematical predictive model along the age-distributed population framework, that can take into account the waning immunity in a transparent and easily controllable manner. Numerical simulations show that under static conditions, the model produces periodic solutions that are qualitatively similar to the reported data, with the period determined by the immunity waning profile. Evidence from the mathematical model indicates that the immunity dynamics is the main factor in the recurrence of infection spikes, however, irregular perturbation of the transmission rate, due to either mutations of the pathogen or human behavior, may result in suppression of recurrent spikes, and irregular time intervals between consecutive peaks. The spike amplitudes are sensitive to the transmission rate and vaccination strategies, but also to the skewness of the profile describing the waning immunity, suggesting that these factors should be taken into consideration when making predictions about future outbreaks.

Modeling surface pH measurements of oocytes.

The transport of gases across cell membranes plays a key role in many different cell functions, from cell respiration to pH control. Mathematical models play a central role in understanding the factors affecting gas transport through membranes, and are the tool needed for testing the novel hypothesis of the preferential crossing through specific gas channels. Since the surface pH of cell membrane is regulated by the transport of gases such as CO2and NH3, inferring the membrane properties can be done indirectly from pH measurements. Numerical simulations based on recent models of the surface pH support the hypothesis that the presence of a measurement device, a liquid-membrane pH sensitive electrode on the cell surface may disturb locally the pH, leading to a systematic bias in the measured values. To take this phenomenon into account, it is necessary to equip the model with a description of the micro-environment created by the pH electrode. In this work we propose a novel, computationally lightweight numerical algorithm to simulate the surface pH data. The effect of different parameters of the model on the output are investigated through a series of numerical experiments with a physical interpretation.

Computational model of electrode-induced microenvironmental effects on pH measurements near a cell membrane.

The mechanism of gas transport across cell membranes remains a topic of considerable interest, particularly regarding the extent to which lipids vs. specific membrane proteins provide conduction pathways. Studies of transmembrane (CO2) transport often rely on data collected under controlled conditions, using pH-sensitive microelectrodes at the extracellular surface to record changes due to extracellular CO2 diffusion and reactions. Although recent detailed computational models can predict a qualitatively correct behavior, a mismatch between the dynamical ranges of the predicted and observed pH curves raises the question whether the discrepancy may be due to a bias introduced by the pH electrode itself. More specifically, it is reasonable to ask whether bringing the electrode tip near or in contact with the membrane creates a local microenvironment between the electrode tip and the membrane, so that the measured data refer to the microenvironment rather than to the free surface. Here, we introduce a detailed computational model, designed to address this question. We find that, as long as a zone of free diffusion exists between the tip and the membrane, the microenvironment behaves effectively as the free membrane. However, according to our model, when the tip contacts the membrane, partial quenching of extracellular diffusion by the electrode rim leads to a significant increase in the pH dynamics under the electrode, matching values measured in physiological experiments. The computational schemes for the model predictions are based on semi-discretization by a finite-element method, and an implicit-explicit time integration scheme to capture the different time scales of the system.

Metabolism plays a central role in the cortical spreading depression: Evidence from a mathematical model.

The slow propagating waves of strong depolarization of neural cells characterizing cortical spreading depression, or depolarization, (SD) are known to break cerebral homeostasis and induce significant hemodynamic and electro-metabolic alterations. Mathematical models of cortical spreading depression found in the literature tend to focus on the changes occurring at the electrophysiological level rather than on the ensuing metabolic changes. In this paper, we propose a novel mathematical model which is able to simulate the coupled electrophysiology and metabolism dynamics of SD events, including the swelling of neurons and astrocytes and the concomitant shrinkage of extracellular space. The simulations show that the metabolic coupling leads to spontaneous repetitions of the SD events, which the electrophysiological model alone is not capable to produce. The model predictions, which corroborate experimental findings from the literature, show a strong disruption in metabolism accompanying each wave of spreading depression in the form of a sharp decrease of glucose and oxygen concentrations, with a simultaneous increase in lactate concentration which, in turn, delays the clearing of excess potassium in extracellular space. Our model suggests that the depletion of glucose and oxygen concentration is more pronounced in astrocyte than neuron, in line with the partitioning of the energetic cost of potassium clearing. The model suggests that the repeated SD events are electro-metabolic oscillations that cannot be explained by the electrophysiology alone. The model highlights the crucial role of astrocytes in cleaning the excess potassium flooding extracellular space during a spreading depression event: further, if the ratio of glial/neuron density increases, the frequency of cortical SD events decreases, and the peak potassium concentration in extracellular space is lower than with equal volume fractions.

Brain energetics plays a key role in the coordination of electrophysiology, metabolism and hemodynamics: Evidence from an integrated computational model.

The energetic needs of brain cells at rest and during elevated neuronal activation has been the topic of many investigations where mathematical models have played a significant role providing a context for the interpretation of experimental findings. A recently proposed mathematical model, comprising a double feedback between cellular metabolism and electrophysiology, sheds light on the interconnections between the electrophysiological details associated with changes in the frequency of neuronal firing and the corresponding metabolic activity. We propose a new extended mathematical model comprising a three-way feedback connecting metabolism, electrophysiology and hemodynamics. Upon specifying the time intervals of higher neuronal activation, the model generates a potassium based signal leading to the concomitant increase in cerebral blood flow with associated vasodilation and metabolic changes needed to sustain the increased energy demand. The predictions of the model are in good qualitative and quantitative agreement with experimental findings reported in the literature, even predicting a slow after-hyperpolarization of a duration of approximately 16 s matching experimental observations.

Estimating hemodynamic stimulus and blood vessel compliance from cerebral blood flow data.

Several key brain imaging modalities that are intended for retrieving information about neuronal activity in brain, the BOLD fMRI as a foremost example, rely on the assumption that elevated neuronal activity elicits spatiotemporally well localized increase of the oxygenated blood volume, which in turn can be monitored non-invasively. The details of the signaling in the neurovascular unit during hyperemia are still not completely understood, and remain a topic of active research, requiring good mathematical models that are able to couple the different aspects of the signaling event. In this work, the question of estimating the hemodynamic stimulus function from cerebral blood flow data is addressed. In the present model, the hemodynamic stimulus is a non-specific signal from the electrophysiological and metabolic complex that controls the compliance of the blood vessels, leading to a vasodilation and thereby to an increase of blood flow. The underlying model is based on earlier literature, and it is further developed in this article for the needs of the inverse problem, which is solved using hierarchical Bayesian methodology, addressing also the poorly known model parameters.

A computational model integrating brain electrophysiology and metabolism highlights the key role of extracellular potassium and oxygen.

The human brain is a small organ which uses a disproportionate amount of the total metabolic energy production in the body. While it is well understood that the most significant energy sink is the maintenance of the neuronal membrane potential during the brain signaling activity, the role of astrocytes in the energy balance continues to be the topic of a lot of research. A key function of astrocytes, besides clearing glutamate from the synaptic clefts, is the potassium clearing after neuronal activation. Extracellular potassium plays a significant role in triggering neuronal firing, and elevated concentration of potassium may lead to abnormal firing patterns, e.g., seizures, thus emphasizing the importance of the glial K+ buffering role. The predictive mathematical model proposed in this paper elucidates the role of glial potassium clearing in brain energy metabolism, integrating a detailed model of the ion dynamics which regulates neuronal firing with a four compartment metabolic model. Because of the very different characteristic time scales of electrophysiology and metabolism, care must be taken when coupling the two models to ensure that the predictions, e.g., neuronal firing frequencies and the oxygen-glucose index (OGI) of the brain during activation and rest, are in agreement with empirical observations. The temporal multi-scale nature of the problem requires the design of new computational tools to ensure a stable and accurate numerical treatment. The model predictions for different protocols, including combinations of elevated activation and ischemic episodes, are in good agreement with experimental observations reported in the literature.

The metabolism of neurons and astrocytes through mathematical models.

Mathematical modeling of the energy metabolism of brain cells plays a central role in understanding data collected with different imaging modalities, and in making predictions based on them. During the last decade, several sophisticated brain metabolism models have appeared. Unfortunately, the picture of the metabolic details that emerges from them is far from coherent: while each model has its justification and is in agreement with some experimental data, some of the predictions of different models can diverge from each other significantly. In this article, we review some of the recent published models, emphasizing similarities and differences between them to understand where the differences in predictions stem from. In that context we present a probabilistic approach, which rather than assigning fixed values to the model parameters, regard them as random variables whose distributions are inferred on in the light of stoichiometric information and different observations. The probabilistic approach reveals how much intrinsic variability a metabolic system may contain, which in turn may be a valid explanation of the different findings.

An adaptive smoothness regularization algorithm for optical tomography.

In diffuse optical tomography (DOT), the object with unknown optical properties is illuminated with near infrared light and the absorption and diffusion coefficient distributions of a body are estimated from the scattering and transmission data. The problem is notoriously ill-posed and complementary information concerning the optical properties needs to be used to counter-effect the ill-posedness. In this article, we propose an adaptive inhomogenous anisotropic smoothness regularization scheme that corresponds to the prior information that the unknown object has a blocky structure. The algorithm updates alternatingly the current estimate and the smoothness penalty functional, and it is demonstrated with simulated data that the algorithm is capable of locating well blocky inclusions. The dynamical range of the reconstruction is improved, compared to traditional smoothness regularization schemes, and the crosstalk between the diffusion and absorption images is clearly less. The algorithm is tested also with a three-dimensional phantom data.

Dynamic Bayesian sensitivity analysis of a myocardial metabolic model.

Dynamic compartmentalized metabolic models are identified by a large number of parameters, several of which are either non-physical or extremely difficult to measure. Typically, the available data and prior information is insufficient to fully identify the system. Since the models are used to predict the behavior of unobserved quantities, it is important to understand how sensitive the output of the system is to perturbations in the poorly identifiable parameters. Classically, it is the goal of sensitivity analysis to asses how much the output changes as a function of the parameters. In the case of dynamic models, the output is a function of time and therefore its sensitivity is a time dependent function. If the output is a differentiable function of the parameters, the sensitivity at one time instance can be computed from its partial derivatives with respect to the parameters. The time course of these partial derivatives describes how the sensitivity varies in time. When the model is not uniquely identifiable, or if the solution of the parameter identification problem is known only approximately, we may have not one, but a distribution of possible parameter values. This is always the case when the parameter identification problem is solved in a statistical framework. In that setting, the proper way to perform sensitivity analysis is to not rely on the values of the sensitivity functions corresponding to a single model, but to consider the distributed nature of the sensitivity functions, inherited from the distribution of the vector of the model parameters. In this paper we propose a methodology for analyzing the sensitivity of dynamic metabolic models which takes into account the variability of the sensitivity over time and across a sample. More specifically, we draw a representative sample from the posterior density of the vector of model parameters, viewed as a random variable. To interpret the output of this doubly varying sensitivity analysis, we propose visualization modalities particularly effective at displaying simultaneously variations over time and across a sample. We perform an analysis of the sensitivity of the concentrations of lactate and glycogen in cytosol, and of ATP, ADP, NAD(+) and NADH in cytosol and mitochondria, to the parameters identifying a three compartment model for myocardial metabolism during ischemia.

Statistical analysis of metabolic pathways of brain metabolism at steady state.

The estimation of metabolic fluxes for brain metabolism is important, among other things, to test the validity of different hypotheses which have been proposed in the literature. The metabolic model that we propose considers, in addition to the blood compartment, the cytosol, and mitochondria of both astrocyte and neuron, including detailed metabolic pathways. In this work we use a recently developed methodology to perform a statistical Flux Balance Analysis (FBA) for this model. The methodology recasts the problem in the form of Bayesian statistical inference and therefore can take advantage of qualitative information about brain metabolism for the simultaneous estimation of all reaction fluxes and transport rates at steady state. By a Markov Chain Monte Carlo (MCMC) sampling method, we are able to provide for each reaction flux and transport rate a distribution of possible values. The analysis of the histograms of the reaction fluxes and transport rates provides a very useful tool for assessing the validity of different hypotheses about brain energetics proposed in the literature, and facilitates the design of the pathways network that is in accordance with what is understood of the functioning of the brain. In this work, we focus on the analysis of biochemical pathways within each cell type (astrocyte and neuron) at different levels of neural activity, and we demonstrate how statistical tools can help implement various bounds suggested by experimental data.

Approximation errors and model reduction in optical tomography.

Model reduction is often required in optical diffusion tomography (ODT), typically due to limited available computation time or computer memory. In practice, this often means that we are bound to use sparse meshes in the model for the forward problem. Conversely, if we are given more and more accurate measurements, we have to employ increasingly accurate forward problem solvers in order to exploit the information in the measurements. In this paper we apply the approximation error theory to ODT. We show that if the approximation errors are estimated and employed, it is possible to use mesh densities that would be unacceptable with a conventional measurement model.

Statistical inversion for medical x-ray tomography with few radiographs: I. General theory.

In x-ray tomography, the structure of a three-dimensional body is reconstructed from a collection of projection images of the body. Medical CT imaging does this using an extensive set of projections from all around the body. However, in many practical imaging situations only a small number of truncated projections are available from a limited angle of view. Three-dimensional imaging using such data is complicated for two reasons: (i) typically, sparse projection data do not contain sufficient information to completely describe the 3D body, and (ii) traditional CT reconstruction algorithms, such as filtered backprojection, do not work well when applied to few irregularly spaced projections. Concerning (i), existing results about the information content of sparse projection data are reviewed and discussed. Concerning (ii), it is shown how Bayesian inversion methods can be used to incorporate a priori information into the reconstruction method, leading to improved image quality over traditional methods. Based on the discussion, a low-dose three-dimensional x-ray imaging modality is described.

Statistical inversion for medical x-ray tomography with few radiographs: II. Application to dental radiology.

Diagnostic and operational tasks in dental radiology often require three-dimensional information that is difficult or impossible to see in a projection image. A CT-scan provides the dentist with comprehensive three-dimensional data. However, often CT-scan is impractical and, instead, only a few projection radiographs with sparsely distributed projection directions are available. Statistical (Bayesian) inversion is well-suited approach for reconstruction from such incomplete data. In statistical inversion, a priori information is used to compensate for the incomplete information of the data. The inverse problem is recast in the form of statistical inference from the posterior probability distribution that is based on statistical models of the projection data and the a priori information of the tissue. In this paper, a statistical model for three-dimensional imaging of dentomaxillofacial structures is proposed. Optimization and MCMC algorithms are implemented for the computation of posterior statistics. Results are given with in vitro projection data that were taken with a commercial intraoral x-ray sensor. Examples include limited-angle tomography and full-angle tomography with sparse projection data. Reconstructions with traditional tomographic reconstruction methods are given as reference for the assessment of the estimates that are based on the statistical model.

Visualization of magnetoencephalographic data using minimum current estimates.

The locations of active brain areas can be estimated from the magnetic field the neural current sources produce. In this work we study a visualization method of magnetoencephalographic data that is based on minimum[symbol: see text] (1)-norm estimates. The method can represent several local or distributed sources and does not need explicit a priori information. We evaluated the performance of the method using simulation studies. In a situation resembling typical magnetoencephalographic measurement, the mean estimated source strength exceeded baseline level up to 2 cm from the simulated point-like source. The method can also visualize several sources, activated simultaneously or in a sequence, which we demonstrated by analyzing magnetic responses associated with sensory stimulation and a picture naming task.

Tikhonov regularization and prior information in electrical impedance tomography.

The solution of impedance distribution in electrical impedance tomography is a nonlinear inverse problem that requires the use of a regularization method. The generalized Tikhonov regularization methods have been popular in the solution of many inverse problems. The regularization matrices that are usually used with the Tikhonov method are more or less ad hoc and the implicit prior assumptions are, thus, in many cases inappropriate. In this paper, we propose an approach to the construction of the regularization matrix that conforms to the prior assumptions on the impedance distribution. The approach is based on the construction of an approximating subspace for the expected impedance distributions. It is shown by simulations that the reconstructions obtained with the proposed method are better than with two other schemes of the same type when the prior is compatible with the true object. On the other hand, when the prior is incompatible with the true object, the method will still give reasonable estimates.