Application of underdamped Langevin dynamics simulations for the study of diffusion from a drug-eluting stent

Regev, Shaked; Farago, Oded

doi:10.1016/j.physa.2018.05.082

Cited by 13 publications

(8 citation statements)

References 43 publications

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“…Finally, as indicated by Lejay [27], since snapping out BM generates sample paths of single-particle diffusion through semipermeable interfaces, it could be used to develop numerical schemes for generating solutions to the corresponding BVP. Indeed, SDEs in the form of underdamped Langevin equations have recently been used to implement efficient computational schemes for finding solutions to the diffusion equation in the presence of one or more semipermeable interfaces [56,57].…”

Section: Discussionmentioning

confidence: 99%

A probabilistic model of diffusion through a semi-permeable barrier

Bressloff

2022

Proc. R. Soc. A.

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Diffusion through semi-permeable structures arises in a wide range of processes in the physical and life sciences. Examples at the microscopic level range from artificial membranes for reverse osmosis to lipid bilayers regulating molecular transport in biological cells to chemical and electrical gap junctions. There are also macroscopic analogues such as animal migration in heterogeneous landscapes. It has recently been shown that one-dimensional diffusion through a barrier with constant permeability κ 0 is equivalent to snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either the left or the right of the barrier. Each round is killed when its Brownian local time exceeds an exponential random variable parameterized by κ 0 . A new round is then immediately started in either direction with equal probability. In this article, we use a combination of renewal theory, Laplace transforms and Green’s function methods to show how an extended version of snapping out BM provides a general probabilistic framework for modelling diffusion through a semi-permeable barrier. This includes modifications of the diffusion process away from the barrier (e.g. stochastic resetting) and non-Markovian models of membrane absorption that kill each round of partially reflected BM. The latter leads to time-dependent permeabilities.

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Section: Discussionmentioning

confidence: 99%

A probabilistic model of diffusion through a semi-permeable barrier

Bressloff

2022

Proc. R. Soc. A.

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“…The mathematical analysis of single-particle diffusion through a semi-permeable interface is important for our understanding of transport phenomena in physical and biological systems. Such processes include molecular transport through lipid bilayers [2,3,4,5], the dynamics of gap junctions [6,7,8], thermal conduction in composite media [9,10,11], diffusion magnetic resonance imaging (dMRI) [12,13,14,15] and drug delivery [16,17,18]. Furthermore, it was recently shown that the trafficking of neurotransmitter receptor proteins in the postsynaptic membrane of neurons can be mathematically formulated in terms of a reaction-diffusion system involving semipermeable membranes which separate the bulk of the neuronal membrane from the synaptic regions [19].…”

Section: Introductionmentioning

confidence: 99%

A numerical method for solving snapping out Brownian motion in 2D bounded domains

Schumm¹,

Bressloff²

2023

Preprint

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Diffusion in heterogeneous media partitioned by semi-permeable interfaces has a wide range of applications in the physical and life sciences, including gas permeation in soils, diffusion magnetic resonance imaging (dMRI), drug delivery, thermal conduction in composite media, synaptic receptor trafficking, and intercellular gap junctions. At the single particle level, diffusion across a semi-permeable interface can be formulated in terms of so-called snapping out Brownian motion (SNOBM). The latter sews together successive rounds of reflected BM, each of which is restricted to one side of the interface. Each round of reflected BM is killed when the local time at the interface exceeds an independent, exponentially distributed random variable. (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of the interface.) The particle then immediately resumes reflected BM on the same side or the other side of the interface according to a stochastic switch, and the process is iterated. In this paper, we develop a Monte Carlo algorithm for simulating a two-dimensional version of SNOBM, which is used to solve a first passage time (FPT) problem for diffusion in a domain with semi-permeable partially absorbing traps. Our method combines a walk-on-spheres (WOS) method with an efficient algorithm for computing the boundary local time that uses a Skorokhod integral representation of the latter. We validate our algorithm by comparing the Monte Carlo estimates of the MFPT to the exact solution for a single circular trap, and show that our simulations are consistent with asymptotic results obtained for the 2D narrow capture problem involving multiple small circular targets. We also discuss extensions to higher dimensions.

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“…As the number of layers and boundaries becomes larger, the calculation of p( r, t) becomes increasingly complicated, which calls for further development of solution methods of diffusion problems in multi-layered systems. In a recent study, we considered the problem of drug release from a drug eluting stent (layer 1) into the artery (layer 2) across a semi-permeable thin membrane (boundary) [16]. We presented a novel approach for finding the time-dependent PDF, which is based on generating a large ensemble of statistically-independent single-particle trajectories using Langevin Dynamics (LD) simulations.…”

Section: Introductionmentioning

confidence: 99%

Algorithms for Brownian dynamics across discontinuities

Farago

2020

Preprint

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The problem of mass diffusion in layered systems has relevance to applications in different scientific disciplines, e.g., chemistry, material science, soil science, and biomedical engineering. The mathematical challenge in these type of model systems is to match the solutions of the time-dependent diffusion equation in each layer, such that the boundary conditions at the interfaces between them are satisfied. As the number of layers increases, the solutions may become increasingly complicated. Here, we describe an alternative computational approach to multi-layer diffusion problems, which is based on the description of the overdamped Brownian motion of particles via the underdamped Langevin equation. In this approach, the probability distribution function is computed from the statistics of an ensemble of independent single particle trajectories. To allow for simulations of Langevin dynamics in layered systems, the numerical integrator must be supplemented with algorithms for the transitions across the discontinuous interfaces. Algorithms for three common types of discontinuities are presented: (i) A discontinuity in the friction coefficient, (ii) a semi-permeable membrane, and (iii) a step-function chemical potential. The general case of an interface where all three discontinuities are present (Kedem-Katchalsky boundary) is also discussed. We demonstrate the validity and accuracy of the derived algorithms by considering a simple two-layer model system and comparing the Langevin dynamics statistics with analytical solutions and alternative computational results.

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Application of underdamped Langevin dynamics simulations for the study of diffusion from a drug-eluting stent

Cited by 13 publications

References 43 publications

A probabilistic model of diffusion through a semi-permeable barrier

A probabilistic model of diffusion through a semi-permeable barrier

A numerical method for solving snapping out Brownian motion in 2D bounded domains

Algorithms for Brownian dynamics across discontinuities

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