Bayesian reconstructions with PDE image model for emission tomography

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.

A closed loop model of pulmonary gas exchange

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.

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.

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).

Modelica is a new object-oriented language for modeling of large, complex, and heterogeneous physical systems. It is designed to allow for convenient and efficient modeling and simulation of the differential-algebraic equations (DAEs) systems. However, in many engineering fields, partial differential equations (PDEs) are applied. So it is necessary to be able to handle PDE models in Modelica. In this paper, we present how the Modelica language can be extended to support object-oriented modeling with partial differential equations. The description of PDE models in Modelica is introduced, which uses a natural way to describe the PDE models so that the extending language is easier to be understood and more convenient to be used. The prototype implementation included external translator and processor is described. An example modeled with PDE in Modelica and solved using the prototype is presented.

We propose an innovative procedure for extracting decay regions from color images of stony materials. The use of appropriate image analysis techniques can offer an important contribution to be used together with the traditional methodologies for studying and diagnosing the decay of stony materials that constitute ancient monuments. The presented approach is constituted by the PDE (Partial Differential Equations) model of anisotropic diffusion and by the level set/fast marching method. The anisotropic diffusion is applied in order to limit the smoothing at the zones of high gradient. In the segmentation process, the numerical technique of the level set/fast marching is applied in order to extract from the image only the color region examined. The study case concerns impressive remains of the city of Aosta (Italy).

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.

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.

Partial differential equations (PDEs) have been successfully applied to many computer vision and image processing problems. However, designing PDEs requires high mathematical skills and good insight into the problems. In this paper, we show that the design of PDEs could be made easier by borrowing the learning strategy from machine learning. In our learning-based PDE (L-PDE) framework for image restoration, there are two terms in our PDE model: (i) a regularizer which encodes the prior knowledge of the image model and (ii) a linear combination of differential invariants, which is data-driven and can effectively adapt to different problems and complex conditions. The L-PDE is learnt from some input/output pairs of training samples via an optimal control technique. The effectiveness of our L-PDE framework for image restoration is demonstrated with two exemplary applications: image denoising and inpainting, where the PDEs are obtained easily and the produced results are comparable to or better than those of traditional PDEs, which were elaborately designed.

Image processing has largely been done via MATLAB and other programming languages, such as JAVA and C, implementing those Partial Differential Equation (PDE) models for denoising, smoothing, segmentation and restoration. This involves significant effort in writing implementation coding and in designing robust algorithms for image processing activities. In order to avoid such repeated effort such as algorithmic development for numerical solutions of PDE, we propose to use existing software packages which provide very good platforms of PDE solvers. Two commercial software platforms, namely PHYSICA3G and ANSYS, are examined. The former one allows a better interaction to the background coding and the latter one does not allow flexible user interaction with the background. This paper describes the implementation of PDE image processing models on and the design of a user interface with two very different software platforms, taking the advantage of the existing PDE solvers within them. The design logic is discussed in the paper along with numerical experiments on a number of methods, including isotropic model and anisotropic model for denoising and smoothing. Several pictures are used in the numerical experiments and the resulting PSNR ratios are obtained and compared with other work in the literature.

This study proposes a new semi-analytical design and implementation method for nonlinear partial differential equation (PDE) control of a flexible manipulator. The proposed scheme considers the effects of the boundary input force and gravity on the payload, which results in non-homogenous boundary conditions. This objective is achieved based on a model transformation scheme for homogenizing boundary conditions, obtaining semi-analytical solutions for the corresponding PDE model. Model transformation is assigned as a hybrid exponential–polynomial function whose coefficients are conveniently calculable without the need for any additional boundary condition measurements. This eliminates the need to use intensive numerical solvers—for example, methods based on finite element analysis—and allows the implementation of sophisticated PDE control schemes considering fully nonlinear PDE models with high computation speed. The presented controller is robust to parametric model uncertainty due to its adaptive design. The precision and efficiency of calculating distributed states using the proposed model transformation are demonstrated based on experimental data for the flexible manipulator with respect to the ground truth camera-based motion capture system. Model transformation is also numerically implemented for the proposed nonlinear endpoint control method based on the original PDE model. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper investigates the difficulty of obtaining data describing the flexible manipulator pose required for precise control and analysis and proposes a computationally efficient method to overcome this issue.