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Coupling volume-excluding compartment-based models of diffusion at different scales: Voronoi and pseudo-compartment approaches.

ABSTRACT: Numerous processes across both the physical and biological sciences are driven by diffusion. Partial differential equations are a popular tool for modelling such phenomena deterministically, but it is often necessary to use stochastic models to accurately capture the behaviour of a system, especially when the number of diffusing particles is low. The stochastic models we consider in this paper are 'compartment-based': the domain is discretized into compartments, and particles can jump between these compartments. Volume-excluding effects (crowding) can be incorporated by blocking movement with some probability. Recent work has established the connection between fine- and coarse-grained models incorporating volume exclusion, but only for uniform lattices. In this paper, we consider non-uniform, hybrid lattices that incorporate both fine- and coarse-grained regions, and present two different approaches to describe the interface of the regions. We test both techniques in a range of scenarios to establish their accuracy, benchmarking against fine-grained models, and show that the hybrid models developed in this paper can be significantly faster to simulate than the fine-grained models in certain situations and are at least as fast otherwise.

SUBMITTER: Taylor PR

PROVIDER: S-EPMC4971222 | biostudies-literature | 2016 Jul

REPOSITORIES: biostudies-literature

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Coupling volume-excluding compartment-based models of diffusion at different scales: Voronoi and pseudo-compartment approaches.

Taylor P R PR Baker R E RE Simpson M J MJ Yates C A CA

Journal of the Royal Society, Interface 20160701 120

Numerous processes across both the physical and biological sciences are driven by diffusion. Partial differential equations are a popular tool for modelling such phenomena deterministically, but it is often necessary to use stochastic models to accurately capture the behaviour of a system, especially when the number of diffusing particles is low. The stochastic models we consider in this paper are 'compartment-based': the domain is discretized into compartments, and particles can jump between th ...[more]

PMID: 27383421

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Project description: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.

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.

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Dataset Information

Coupling volume-excluding compartment-based models of diffusion at different scales: Voronoi and pseudo-compartment approaches.

Publications

Coupling volume-excluding compartment-based models of diffusion at different scales: Voronoi and pseudo-compartment approaches.

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