Fixed Boundary PDE Model Output

  • William E. Schiesser


The numerical and graphical output from execution of the R routines in Chap.  2 (fixed tumor outer boundary) is discussed in this chapter. Specifically, five cases are considered: (1) no cancer vaccine (CV), no immune checkpoint inhibitor (ICI), (2) CV added, no ICI, (3) no CV, ICI added, (4) CV, ICI added, (5) CV, ICI added, higher dosage. Of particular interest in the output is the density (concentration) of cancer cells. For cases (1) to (4), the cancer cells remain for increasing time. For case (5) a cancer-free condition evolves for increasing time, thus indicating that increasing the CV-ICI dosages above the base level of case (4) gives an effective cancer therapy.


Cancer vaccine (CV) dosage Immune checkpoint inhibitor (ICI) dosage Mathematical model output Computer-based experimentation Cancer-free outcome 

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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