Moving Boundary PDE Model Implementation

  • William E. Schiesser


The partial differential equation (PDE) formulation in Chap.  1 and implementation in Chap.  2 for a fixed tumor outer boundary is extended in this chapter to a moving boundary by the use an algorithm based on an equation for the outer boundary velocity. Three cases for the outer boundary velocity are considered: (1) a fixed boundary (zero velocity) for comparison with the results discussed in Chap.  3, (2) a constant velocity which can be checked (the outer boundary moves linearly in time), and (3) the outer boundary velocity is proportional to the cancer density at the outer boundary. For the three cases, the plotted output for the outer boundary velocity and position are of particular interest.


Partial differential equation (PDE) Mathematical model for moving boundary (MBPDE) Algorithm for moving boundary Boundary velocity equation Boundary cancer cell density R main program implementation 

Supplementary material


  1. 1.
    Lai, X., and A. Friedman. 2017. Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: A mathematical model. PLoS One 12(5):e0178479.CrossRefGoogle Scholar
  2. 2.
    Schiesser, W.E. 2017. Spline collocation methods for partial differential equations: With applications in R. Hoboken: Wiley.CrossRefGoogle Scholar
  3. 3.
    Soetaert, K., J. Cash, and F. Mazzia. 2012. Solving differential equations in R. Heidelberg: Springer.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • William E. Schiesser
    • 1
  1. 1.Department of Chemical and Biomolecular EngineeringLehigh UniversityBethlehemUSA

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