Abstract
The output from the moving boundary partial differential equation (MBPDE) algorithm discussed in Chap. 4 is presented in this chapter. First, the velocity and position of the tumor outer boundary are confirmed for the two test cases explained in Chap. 4 (zero and constant boundary velocities). Then the velocity and position of the boundary are presented for the third case of the velocity proportional to the boundary cancer cell density. In particular, the effect of the proportionality constant in the velocity equation of the third case is demonstrated. As in case (5) of Chap. 3, a cancer-free condition evolves for increasing time, thus indicating that increasing the CV-ICI dosages gives an effective cancer therapy.
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Notes
- 1.
Each call to lsodes requires a start with the first order implicit Euler method, then an automatic order increase in the integration algorithm (based on backward differentiation formulas, i.e., the BDF method [1]) to possibly a maximum order of five. This process of order variation requires substantial numerical calculations as reflected in the value of ncall.
- 2.
The solution of the 12 PDE model is essentially invariant with movement of the outer radial boundary. This property follows from the use of integral average cell densities and biochemical concentrations. The solution before r refinement is the IC for the solution after r refinement. This procedure based on the integral average gives variation in t that does not depend on the refined r. Investigation of the solution radial profiles should provide elucidation of this solution property. The radial profiles can be readily plotted from the computed solutions, and this is left as an exercise.
References
Soetaert, K., J. Cash, and F. Mazzia. 2012. Solving differential equations in R. Heidelberg: Springer.
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Schiesser, W.E. (2019). Moving Boundary PDE Model Output. In: Spatiotemporal Modeling of Cancer Immunotherapy. Springer, Cham. https://doi.org/10.1007/978-3-030-19080-4_5
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DOI: https://doi.org/10.1007/978-3-030-19080-4_5
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-17635-8
Online ISBN: 978-3-030-19080-4
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