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The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion.

ABSTRACT: Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, owing to recent advances in computational power, the simulation and therefore postulation, of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction-diffusion systems in which regions of low copy numbers exist. The specific stochastic models with which we shall be concerned in this manuscript are referred to as 'compartment-based' or 'on-lattice'. These models are characterized by a discretization of the computational domain into a grid/lattice of 'compartments'. Within each compartment, particles are assumed to be well mixed and are permitted to react with other particles within their compartment or to transfer between neighbouring compartments. Stochastic models provide accuracy, but at the cost of significant computational resources. For models that have regions of both low and high concentrations, it is often desirable, for reasons of efficiency, to employ coupled multi-scale modelling paradigms. In this work, we develop two hybrid algorithms in which a PDE in one region of the domain is coupled to a compartment-based model in the other. Rather than attempting to balance average fluxes, our algorithms answer a more fundamental question: 'how are individual particles transported between the vastly different model descriptions?' First, we present an algorithm derived by carefully redefining the continuous PDE concentration as a probability distribution. While this first algorithm shows very strong convergence to analytical solutions of test problems, it can be cumbersome to simulate. Our second algorithm is a simplified and more efficient implementation of the first, it is derived in the continuum limit over the PDE region alone. We test our hybrid methods for functionality and accuracy in a variety of different scenarios by comparing the averaged simulations with analytical solutions of PDEs for mean concentrations.

SUBMITTER: Yates CA

PROVIDER: S-EPMC4424691 | biostudies-literature | 2015 May

REPOSITORIES: biostudies-literature

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The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion.

Yates Christian A CA Flegg Mark B MB

Journal of the Royal Society, Interface 20150501 106

Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, owing to recent advances in computational power, the simulation and therefore postulation, of computationally intensive individual-based models has become ...[more]

PMID: 25904527

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Project description:Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

Project description:OBJECTIVE:Focal cortical dysplasia (FCD) lesion detection and subtyping remain challenging on conventional MRI. New diffusion models such as the spherical mean technique (SMT) and neurite orientation dispersion and density imaging (NODDI) provide measurements that potentially produce more specific maps of abnormal tissue microstructure. This study aims to assess the SMT and NODDI maps for computational and radiological lesion characterization compared to standard fractional anisotropy (FA) and mean diffusivity (MD). METHODS:SMT, NODDI, FA, and MD maps were calculated for 33 pediatric patients with suspected FCD (18 histologically confirmed). Two neuroradiologists scored lesion visibility on clinical images and diffusion maps. Signal profile changes within lesions and homologous regions were quantified using a surface-based approach. Diffusion parameter changes at multiple cortical depths were statistically compared between FCD type IIa and type IIb. RESULTS:Compared to fluid-attenuated inversion recovery (FLAIR) or T1-weighted imaging, lesions conspicuity on NODDI intracellular volume fraction (ICVF) maps was better/equal/worse in 5/14/14 patients, respectively, while on SMT intra-neurite volume fraction (INVF) in 3/3/27. Compared to FA or MD, lesion conspicuity on the ICVF was better/equal/worse in 27/4/2, while on the INVF in 20/7/6. Quantitative signal profiling demonstrated significant ICVF and INVF reductions in the lesions, whereas SMT microscopic mean, radial, and axial diffusivities were significantly increased. FCD type IIb exhibited greater changes than FCD type IIa. No changes were detected on FA or MD profiles. SIGNIFICANCE:FCD lesion-specific signal changes were found in ICVF and INVF but not in FA and MD maps. ICVF and INVF showed greater contrast than FLAIR in some cases and had consistent signal changes specific to FCD, suggesting that they could improve current presurgical pediatric epilepsy imaging protocols and can provide features useful for automated lesion detection.

Dataset Information

The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion.

Publications

The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion.

Similar Datasets

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