A closed loop model of pulmonary gas exchange

The effect of blood flow and diffusion impairment on pulmonary gas exchange: A computer model

Experimental results have shown that anti-cancer therapies, such as radiotherapy and chemotherapy, can modulate the cell cycle and generate cell cycle phase-dependent responses. As a result, obtaining a detailed understanding of the cell cycle is one possible path towards improving the efficacy of many of these therapies. Here, we consider a basic structured partial differential equation (PDE) model for cell progression through the cell cycle, and derive expressions for key quantities, such as the population growth rate and cell phase proportions. These quantities are shown to be periodic and, as such, we compare the PDE model to a corresponding ordinary differential equation (ODE) model in which the parameters are linked by ensuring that the long-term ODE behaviour agrees with the average PDE behaviour. By design, we find that the ODE model does an excellent job of representing the mean dynamics of the PDE model within just a few cell cycles. However, by probing the parameter space we find cases in which this mean behaviour is not a good measure of the PDE population growth. Our analytical comparison of two caricature models (one PDE and one ODE system) provides insight into cases in which the simple ODE model is an appropriate approximation to the PDE model.

IntroductionAlveolar dead space reflects phenomena that render arterial partial pressure of carbon dioxide higher than that of mixed alveolar gas, disturbing carbon dioxide exchange. Right-to-left shunt fraction (Qs/Qt) leads to an alveolar dead space fraction (VdAS/VtA; where VtA is alveolar tidal volume). In acute respiratory distress syndrome, ancillary physiological disturbances may include low cardiac output, high metabolic rate, anaemia and acid-base instability. The purpose of the present study was to analyze the extent to which shunt contributes to alveolar dead space and perturbs carbon dioxide exchange in ancillary physiological disturbances.MethodsA comprehensive model of pulmonary gas exchange was based upon known equations and iterative mathematics.ResultsThe alveolar dead space fraction caused by shunt increased nonlinearly with Qs/Qt and, under 'basal conditions', reached 0.21 at a Qs/Qt of 0.6. At a Qs/Qt of 0.4, reduction in cardiac output from 5 l/minute to 3 l/minute increased VdAS/VtA from 0.11 to 0.16. Metabolic acidosis further augmented the effects of shunt on VdAS/VtA, particularly with hyperventilation. A Qs/Qt of 0.5 may increase arterial carbon dioxide tension by about 15% to 30% if ventilation is not increased.ConclusionIn acute respiratory distress syndrome, perturbation of carbon dioxide exchange caused by shunt is enhanced by ancillary disturbances such as low cardiac output, anaemia, metabolic acidosis and hyperventilation. Maintained homeostasis mitigates the effects of shunt.

In this study, problems of the partial differential equation (PDE) modelling and vibration control design are first resolved for an overhead crane bridge system, which is regarded as a Timoshenko beam with an attached rigid body. With the established infinite‐dimensional PDE model, a Nussbaum function‐based adaptive control approach is developed in order to settle out parametric uncertainties of the system with unknown control directions. Under the proposed control method, states of the overhead crane bridge are globally bounded and finally converge to an adjustable region of zero. The stability of the closed‐loop overhead crane bridge system is analysed by employing Lyapunov's direct method. The effectiveness of the developed control strategy is verified through the simulation results.

Estimating the Measuring Effort for Using a Two-Parameter Gas Exchange Model in Clinical Practice

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat "patchy" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice-directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs.

To model biochemical systems in which both noise in the chemical reaction process and spatial movement of molecules is important, both the reaction-diffusion master equation (RDME) and Smoluchowski diffusion-limited reaction (SDLR) partial differential equation (PDE) models have been used. In previous work we showed that the solution to the RDME may be interpreted as an asymptotic approximation in the reaction radius to the solution of the SDLR PDE [S. A. Isaacson, SIAM J. Appl. Math. 70, 77 (2009)]. The approximation was shown to be divergent in the limit that the lattice spacing in the RDME approached zero. In this work we expand upon these results for the special case of the two-molecule annihilation reaction, A+B-->Ø. We first introduce a third stochastic reaction-diffusion PDE model that incorporates a pseudopotential based bimolecular reaction mechanism. The solution to the pseudopotential model is then shown to be an asymptotic approximation to the solution of the SDLR PDE for small reaction radii. We next illustrate how the RDME may be obtained by a formal discretization of the pseudopotential model, motivating why the RDME is itself an asymptotic approximation of the SDLR PDE. Finally, we give a more detailed numerical analysis of the difference between solutions to the RDME and SDLR PDE models as a function of both the reaction-radius and the lattice spacing (in the RDME).

Summary There is no standard framework for solving optimization problems for systems described by agent-based models (ABMs). We present a method for constructing individual-level controls that steer the population-level dynamics of an ABM towards a desired state. Our method uses a system of partial differential equations (PDEs) with control functions to approximate the dynamics of the ABM with control. An optimal control problem is formulated in terms of the PDE model to mimic the optimization goal of the ABM. Mathematical theory is used to derive optimal controls for the PDE model, which are numerically approximated and transformed for use in the ABM. We use the Sugarscape ABM, a prototype ABM that includes agent and environmental heterogeneity and accumulation of agent resources over time. We present a PDE model that approximates well the spatial, temporal, and resource dynamics of the Sugarscape ABM. In both models, control represents taxation of agent wealth with the goal to maximize total taxes collected while minimizing the impact of taxation on the population over a finite time. Solutions to the optimal control problem yield taxation rates specific to an agent's location and current wealth. The use of optimal controls (generated by the PDE model) within the ABM performed better than other controls we evaluated, even though some error was introduced between the ABM and PDE models. Our results demonstrate the feasibility of using a PDE to approximate an ABM for control purposes and illustrate challenges that can arise in applying this technique to sophisticated ABMs. Copyright © 2016 John Wiley & Sons, Ltd.

The essential function of the lung, gas exchange, is dependent on adequate matching of ventilation and perfusion, where air and blood are delivered through complex branching systems exposed to regionally varying transpulmonary and transmural pressures. Structure and function in the lung are intimately related, yet computational models in pulmonary physiology usually simplify or neglect structure. The geometries of the airway and vascular systems and their interaction with parenchymal tissue have an important bearing on regional distributions of air and blood, and therefore on whole lung gas exchange, but this has not yet been addressed by modelling studies. Models for gas exchange have typically incorporated considerable detail at the level of chemical reactions, with little thought for the influence of structure. To date, relatively little attention has been paid to modelling at the cellular or subcellular level in the lung, or to linking information from the protein structure/interaction and cellular levels to the operation of the whole lung. We review previous work in developing anatomically based models of the lung, airways, parenchyma and pulmonary vasculature, and some functional studies in which these models have been used. Models for gas exchange at several spatial scales are briefly reviewed, and the challenges and benefits from modelling cellular function in the lung are discussed.

Accurate reporting and forecasting of PM2.5 concentration are important for improving public health. In this paper, we propose a partial differential equation (PDE) model, specially, a linear diffusive equation, to describe the spatial-temporal characteristics of PM2.5 in order to make short-term prediction. We analyze the temporal and spatial patterns of a real dataset from China’s National Environmental Monitoring and validate the PDE-based model in terms of predicting the PM2.5 concentration of the next day by the former days’ history data. Our experiment results show that the PDE model is able to characterize and predict the process of PM2.5 transport. For example, for 300 continuous days of 2016, the average prediction accuracy of the PDE model over all city-regions is 93% or 83% based on different accuracy definitions. To our knowledge, this is the first attempt to use PDE-based model to study PM2.5 prediction in both temporal and spatial dimensions.

A partial differential equation (PDE) model for a flexible inverted pendulum system (FIPS) is derived by the use of the Hamilton principle. To solve the coupling system model, a singular perturbation method was adopted. The PDE model was divided into a fast subsystem and a slow subsystem using the singular perturbation method. To stabilize the fast subsystem, a boundary control force was applied at the free end of the beam. It then was proven that the closed‐loop subsystem is appropriate and exponentially stable. For the slow subsystem, a sliding mode control method was employed to design a controller and the Linear Matrix Inequality (LMI) method was used to design the sliding surface. It then was shown that the slow subsystem is exponentially stable.

Populations interact in a wide variety of ways: through cooperation,competition, or predation. In this work we consider two interacting populations, individuals and a stimulus. Individuals move in response to the stimulus population while the stimulus only diffuses. Both populations grow while the stimulus population is being depleted by the individuals. In order to account for the random nature of the system, an individual-based model (IBM) is first developed and then upscaled into a continuous partial differential equation (PDE) model by considering transition probabilities of the individuals at each site. Finally, a set of numerical experiments is presented showing very good agreement between the IBM and the PDE model.

Division of Surgical Intensive Care, Department of Anesthesiology, Pharmacology, and Surgical Intensive Care (APSIC), University Hospital of Geneva, Geneva, Switzerland. peter.suter@hcuge.chACUTE lung injury (ALI) and adult respiratory distress syndrome (ARDS) are characterized by the tendency of peripheral air spaces to collapse, particularly in dependent lung areas. The application of a positive end-expiratory pressure (PEEP) and an appropriate tidal volume (VT) is thought to prevent and reverse the collapse of airways and respiratory units in such patients. Other means to recruit alveoli include frequent position changes, e.g . from supine to prone. The introduction of specific inflation maneuvers, using either an intermittent high-tidal volume, or an elevated inspiratory (plateau) pressure, allows significant increases in end-inspiratory and end-expiratory lung volume, and an associated improvement in gas exchange. Clinical results obtained with such recruitment strategies are reported in the current issue of Anesthesiology by two groups of well-known clinical investigators. 1,2Both articles describe the effects of lung inflation in patients with ARDS, studied in the early or late phases of their disease. The advent of ventilatory techniques using lower VTto decrease ventilator-induced lung injury and improve outcome in ALI and ARDS 3,4has renewed interest in the requirement for, and the potential benefit of, periodic inflation maneuvers. It has indeed been shown that VTparticipates to a significant extent in the opening of collapsed or hypoventilated areas in such patients. 5Therefore, it is not surprising that with the new low VTapproach, an old intermittent inflation method, called sigh, 6would regain some credibility in ventilatory management. In the first of the two studies discussed here, Salvatore Grasso et al . reports the effects of a continuous 40-cm H2O inflation pressure applied for 3–5 s in 22 patients undergoing controlled mechanical ventilation (MV) with a VTof 6 ml/kg and a mean PEEP of 9 cm H2O. Pulmonary gas exchange, respiratory mechanics, and hemodynamics were assessed. The investigation was done at two different times during the evolution of ARDS, i.e. in early (1 to 2 days after beginning MV) or later phases (5–10 days after onset of MV). The results are clearcut: only in the early phase of the disease was a marked positive response in the Pao2/ Fio2ratio observed. Patients in the later phases of ARDS had smaller increases in oxygenation, which were associated with a significant decrease in cardiac output during the inflation maneuver. This difference may simply be related to respiratory mechanics; in the early phase of ARDS, the elastance of the lung and chest wall were lower, which means higher compliance values, and hence, larger increases in volume for a given change in distending pressure.In the second study, 2Nicolo Patroniti et al. describes thirteen ARDS patients intubated and breathing spontaneously, assisted by an inspiratory pressure support of 12 cm H2O and a PEEP of 11 cm H2O, resulting in a VTof 420 ml. A majority of these patients can be considered to be in an early phase of the disease. The recruitment maneuver applied was similar to that used by Patroniti et al. ; airway pressure was increased to a mean of 38 cm H2O for 3.6 s, resulting in a VTof 1,150 ml. This sigh produced marked improvements in gas exchange and expiratory lung volume and a decrease in respiratory drive. What can we learn from these two elegant investigations? First, the findings confirm that sustained inflation using a pressure of 30 to 45 cm H2O results in a marked improvement in arterial oxygenation and has no significant side effects. 7Second, the reader will not be surprised to see that in early ARDS, functional respiratory units can be inflated and recruited more easily than in later phases of the disease, and that the gain in arterial oxygenation is more impressive. Similar results have been shown for prone positioning where improvement in gas exchange is greater in patients with pulmonary edema or with early ARDS, as compared with late ARDS or pulmonary fibrosis. 8Third, it should not be forgotten that alveolar recruitment and pulmonary gas exchange depend largely on the level of PEEP applied during protective ventilatory strategies. 9In conclusion, the two articles on recruitment maneuvers published in this issue add important information for the clinician; however, a number of questions remain. For instance, further studies should explore which single or combined therapy, including higher PEEP levels, regular position changes, and intermittent inflation strategies achieves the greatest sustained improvement in lung function. In addition, the effects on outcome in ARDS patients, including end-points such as duration of mechanical ventilation, duration of stay in the intensive care unit, and hospital mortality, must be assessed. It seems likely that understanding the role of factors such as the underlying morphologic features of lung injury, the stage of ARDS, hemodynamic status, and tolerance to increased intrapulmonary pressure will be essential to determining optimal treatment strategies for individual patients.

The aim of the present study was to investigate a new type of Bayesian reconstruction method which utilizes partial differential equations (PDE) image models as a prior. PDE image models are very popular in image restoration and segmentation. Our method introduces such models to emission tomography reconstruction by using Bayesian one step late (OSL) algorithm and an OS acceleration one. In a PDE based model, the image can be viewed as the solution of an evolutionary differential equation. It can be thought as a descent of an energy function which entitled us to use PDE models in Bayesian reconstruction. In this paper, three different PDE models are studied, two of them are based on anisotropic diffusion model, and another is based on a complex diffusion model. All of them have the properties of edge-preserving and denoising as the latest median root prior (MRP). We validated the effectiveness of the method using a Zubal phantom in numerical experiments and compared it to the classical MLEM and MRP reconstruction. The results show that the proposed PDE model method is better than the MLEM and the MRP reconstruction methods in visualization, bias and variance, and are more suitable for OS acceleration than MRP.