# American Institute of Mathematical Sciences

January  2017, 22(1): 115-123. doi: 10.3934/dcdsb.2017006

## Optimisation modelling of cancer growth

 Western Australian Centre of Excellence in Industrial Optimisation, Curtin University, Perth, 6102, Australia and Department of Mathematics and Statistics, Curtin University, Perth, 6102, Australia

* Corresponding author: Tiffany A. Jones

Received  January 2016 Revised  June 2016 Published  December 2016

Several computational models have been developed in the literature to describe the dynamics of the cell-cycle for the mammalian cell, in particular for cancer cells, using both traditional and new techniques and yielding some positive results. In this paper, we discuss how to optimise model parameters for these types of models and how this can serve to enhance numerical results. We pose the model parameter selection problem as an optimal parameter selection problem on both a normal cell and cancer cell cycle of growth. Various possible objectives are discussed and we illustrate the process with some numerical results.

Citation: Tiffany A. Jones, Lou Caccetta, Volker Rehbock. Optimisation modelling of cancer growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 115-123. doi: 10.3934/dcdsb.2017006
##### References:

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##### References:
Results from MISER3.3 for Multiple Characteristic Time points
 $\mathbf{\zeta}$ Initial Value MISER3.3 Rounded Tyson $\zeta_{1}$ 0.04 0.048528 0.05 0.04 $\zeta_{2}$ 0.04 0.0420341 0.04 0.04 $\zeta_{3}$ 1.5 1.00497 1.0 1.0 $\zeta_{4}$ 1.5 1.09633 1.1 1.0 $\zeta_{5}$ 12.0 9.74194 10.0 10.0 $\zeta_{6}$ 35.0 35.7286 36.0 35.0 $\zeta_{7}$ 0.005 0.00490765 0.005 0.005 $\zeta_{8}$ 0.2 0.199762 0.2 0.2 $\zeta_{9}$ 0.1 0.099762 0.1 0.1 $\zeta_{10}$ 1.2 1.00158 1.0 1.0 $\zeta_{11}$ 0.6 0.502537 0.5 0.5 $\zeta_{12}$ 0.2 0.101318 0.1 0.1 $\zeta_{13}$ 0.01 0.020161 0.02 0.02 $\zeta_{14}$ 0.01 0.009979 0.01 0.01
 $\mathbf{\zeta}$ Initial Value MISER3.3 Rounded Tyson $\zeta_{1}$ 0.04 0.048528 0.05 0.04 $\zeta_{2}$ 0.04 0.0420341 0.04 0.04 $\zeta_{3}$ 1.5 1.00497 1.0 1.0 $\zeta_{4}$ 1.5 1.09633 1.1 1.0 $\zeta_{5}$ 12.0 9.74194 10.0 10.0 $\zeta_{6}$ 35.0 35.7286 36.0 35.0 $\zeta_{7}$ 0.005 0.00490765 0.005 0.005 $\zeta_{8}$ 0.2 0.199762 0.2 0.2 $\zeta_{9}$ 0.1 0.099762 0.1 0.1 $\zeta_{10}$ 1.2 1.00158 1.0 1.0 $\zeta_{11}$ 0.6 0.502537 0.5 0.5 $\zeta_{12}$ 0.2 0.101318 0.1 0.1 $\zeta_{13}$ 0.01 0.020161 0.02 0.02 $\zeta_{14}$ 0.01 0.009979 0.01 0.01
Table of $\tau_{i}$ values for characteristic time points
 $\begin{gathered} \overline {\underline {{\tau _i}\;\;{\text{Times}}} } \hfill \\ {\tau _1}\;\;0 \hfill \\ {\tau _2}\;\;142 \hfill \\ {\tau _3}\;\;284 \hfill \\ {\tau _4}\;\;426 \hfill \\ {\tau _5}\;\;568 \hfill \\ {\tau _6}\;\;710 \hfill \\ {\tau _7}\;\;852 \hfill \\ {\tau _8}\;\;994 \hfill \\ {\tau _9}\;\;1136 \hfill \\ {\tau _{10}}\;\;1278 \hfill \\ \underline {{\tau _{11}}\;\;1420} \hfill \\ \end{gathered}$
 $\begin{gathered} \overline {\underline {{\tau _i}\;\;{\text{Times}}} } \hfill \\ {\tau _1}\;\;0 \hfill \\ {\tau _2}\;\;142 \hfill \\ {\tau _3}\;\;284 \hfill \\ {\tau _4}\;\;426 \hfill \\ {\tau _5}\;\;568 \hfill \\ {\tau _6}\;\;710 \hfill \\ {\tau _7}\;\;852 \hfill \\ {\tau _8}\;\;994 \hfill \\ {\tau _9}\;\;1136 \hfill \\ {\tau _{10}}\;\;1278 \hfill \\ \underline {{\tau _{11}}\;\;1420} \hfill \\ \end{gathered}$
Table of simulated parameter values
 $\mathbf{\zeta}$ MISER3.3 Rounded Alarcón $\zeta_{1}$ 0.00864931 0.009 0.04 $\zeta_{2}$ 0.990733 1.0 1.0 $\zeta_{3}$ 0.330044 0.33 0.25 $\zeta_{4}$ 0.0807658 0.08 0.04 $\zeta_{5}$ 10.0014 10.0 10.0 $\zeta_{6}$ 3.49825 3.5 3.5 $\zeta_{7}$ 0.0966571 0.1 0.01 $\zeta_{8}$ 0.07 0.07 0.007 $\zeta_{9}$ 0.0195864 0.02 0.01 $\zeta_{10}$ 0.00721735 0.007 0.01 $\zeta_{11}$ 0.000292239 0.0003 0.01 $\zeta_{12}$ 0.00494411 0.005 0.1
 $\mathbf{\zeta}$ MISER3.3 Rounded Alarcón $\zeta_{1}$ 0.00864931 0.009 0.04 $\zeta_{2}$ 0.990733 1.0 1.0 $\zeta_{3}$ 0.330044 0.33 0.25 $\zeta_{4}$ 0.0807658 0.08 0.04 $\zeta_{5}$ 10.0014 10.0 10.0 $\zeta_{6}$ 3.49825 3.5 3.5 $\zeta_{7}$ 0.0966571 0.1 0.01 $\zeta_{8}$ 0.07 0.07 0.007 $\zeta_{9}$ 0.0195864 0.02 0.01 $\zeta_{10}$ 0.00721735 0.007 0.01 $\zeta_{11}$ 0.000292239 0.0003 0.01 $\zeta_{12}$ 0.00494411 0.005 0.1
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