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A New Fick-Jacobs Derivation with Applications to Computational Branched Diffusion Networks

Zachary M. Miksis, Gillian Queisser

TL;DR

This work tackles diffusion in tubular and branched networks where the classical $Fick-Jacobs$ reduction can be unstable under steep radial gradients. It introduces Extended-Flux Fick-Jacobs (EF-FJ), incorporating higher-order spatial derivatives to stabilize the reduced model and improve convergence, together with simple, conserved discretizations for branching points. Through rigorous stability analysis and extensive numerical experiments on cones, sinusoidal channels, and branched domains, EF-FJ demonstrates superior accuracy and efficiency over traditional corrections, including in a neuroscience-inspired intracellular-calcium diffusion scenario. The results establish EF-FJ as a robust, dimension-reduced framework for fast, accurate diffusion simulations in complex branched geometries with practical applications in biology and neuroscience.

Abstract

The Fick-Jacobs equation is a classical model reduction of 3-dimensional diffusion in a tube of varying radius to a 1-dimensional problem with radially scaled derivatives. This model has been shown to be unstable when the radial gradient is too steep. In this work, we present a new derivation of the Fick-Jacobs equation that results in the addition of higher order spatial derivative terms that provide additional stability in a wide variety of cases and improved solution convergence. We also derive new numerical schemes for branched nodes within networks and provide stability conditions for these schemes. The computational accuracy, efficiency, and stability of our method is demonstrated through a variety of numerical examples.

A New Fick-Jacobs Derivation with Applications to Computational Branched Diffusion Networks

TL;DR

This work tackles diffusion in tubular and branched networks where the classical reduction can be unstable under steep radial gradients. It introduces Extended-Flux Fick-Jacobs (EF-FJ), incorporating higher-order spatial derivatives to stabilize the reduced model and improve convergence, together with simple, conserved discretizations for branching points. Through rigorous stability analysis and extensive numerical experiments on cones, sinusoidal channels, and branched domains, EF-FJ demonstrates superior accuracy and efficiency over traditional corrections, including in a neuroscience-inspired intracellular-calcium diffusion scenario. The results establish EF-FJ as a robust, dimension-reduced framework for fast, accurate diffusion simulations in complex branched geometries with practical applications in biology and neuroscience.

Abstract

The Fick-Jacobs equation is a classical model reduction of 3-dimensional diffusion in a tube of varying radius to a 1-dimensional problem with radially scaled derivatives. This model has been shown to be unstable when the radial gradient is too steep. In this work, we present a new derivation of the Fick-Jacobs equation that results in the addition of higher order spatial derivative terms that provide additional stability in a wide variety of cases and improved solution convergence. We also derive new numerical schemes for branched nodes within networks and provide stability conditions for these schemes. The computational accuracy, efficiency, and stability of our method is demonstrated through a variety of numerical examples.
Paper Structure (23 sections, 2 theorems, 52 equations, 15 figures, 3 tables)

This paper contains 23 sections, 2 theorems, 52 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Expanded Flux Fick-Jacobs Derivation
  3. Original Fick-Jacobs Derivation
  4. Diffusion Coefficient Correction
  5. Kalinay Temporal Correction
  6. Expanded Flux Fick-Jacobs Equation
  7. Derivation of the Expanded Flux Fick-Jacobs equation
  8. Branch Point Conservation Conditions
  9. Numerical Methods
  10. Boundary Nodes
  11. Stability Analysis
  12. Numerical Results
  13. Truncated Cone
  14. Results
  15. Temporal Stability
  16. ...and 8 more sections

Key Result

Theorem 1

Given the diffusion equation the second order branched forward-time central-space (FTCS) scheme is stable given the condition is satisfied.

Figures (15)

  • Figure 1: Jacobs flux through a truncated cone.
  • Figure 2: Expanded Jacobs flux through a truncated cone.
  • Figure 3: A discretized branch point of a network.
  • Figure 4: Possible branching 2-paths $P_i$ with concentrations $c_i$.
  • Figure 5: Numerical approximations by different models with exact solution at $t = 10$ in a truncated cone.
  • ...and 10 more figures

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1: Diffusion is stable at branch nodes
  • proof
  • Theorem 2: First order term is stable at branch nodes
  • proof