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Cardiac Physiome Society workshop: November 6-9, 2017 , Toronto

# DiffusionTutorial

Several approaches to modeling diffusion in combination with other processes are illustrated.

## Description

```The following approaches are detailed:

1-D Diffusion modeled as a partial differential equation: Model # 0330
Partial differential equation with no flux boundary conditions (Neumann),
initialized with a centered spike.

1-D Diffusion with asymmetrical Consumption modeled as a partial differential equation: Model # 0364
Same as 1-D Diffusion with consumption of metabolite as a function of x.

1-D diffusion-advection equation with Robin boundary condition: Model # 0169
Similar to 1-D diffusion with added advection term. Initial value is zero. There is
an inflow concentration, Cin. Uses a Robin condition for inflow boundary condition.

Random Walks of multiple particles in 1 dimension: Model # 0184
Multiple realizations of 1-D random walks. Final positions are binned and compared to
theoretical calculations.

Random Walk of single particle in 2 dimensions: Model # 0372
A single random walk in two dimensions is plotted wth with green circles, marked with
starts every nsteps. Three kinds of steps may be chosen:
(1) fixed step sizes
(2) random step sizes
(3) random step size with random angle.

Fractional Brownian Motion Walk in 2 dimensions: Model # 0374
Uses increments of fractional Gaussian noise to create fractional Brownian Motion
using the Davies-Harte algorithm. (See FGP model for details.) Illustrates that
steps can be Gaussian, but the Hurst Coefficient measuring correlation of steps
is highly important.

Diffusion in a uniform slab: Model # 176
Similar to 1-D Diffusion with addition of a partition coefficient lambda,
the ratio of the concentration immediately inside the region to that outside.
Uses Dirichlet boundary conditions.

Diffusion in two uniform slabs with different diffusivities: Model 0212
Similar to Diffusion in a uniform slab, but with the diffusion coefficient
containing a discontinuity and the boundary of two media.

Laplace's equation in 2-D with Dirichlet Boundary conditions: Model # 0363
Laplace's equation in two dimension is solved using ordinary different equations
and solved again used 1-d partial differential equations. both methods use
second order accurate finite difference approximations.

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## Key Terms

diffusion tutorial, diffusion, tutorial, 1d, 1-d, 1D, 1_d, one dimension, 2d, 2-d, 2D, 2-D, two dimension, PDE, Dirichlet, Neumann, Robin, boundary condition, Random Walk, fractional Brownian motion, fBm, fGn, FGP

## Model History

Get Model history in CVS.

Posted by: GR

## Acknowledgements

Please cite www.physiome.org in any publication for which this software is used and send an email with the citation and, if possible, a PDF file of the paper to: staff@physiome.org.
Or send a copy to:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.