# 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 and Continuous Dynamical Systems - B, 2017, 22 (1) : 115-123. doi: 10.3934/dcdsb.2017006
##### References:
 [1] T. Alarcón, H. M. Byrne and P. K. Maini, A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells, JTB, 229 (2004), 397-411. [2] S. Dormann and A. Deutsch, Modeling of self-organized avascular tumor growth with a hybrid cellular automaton, In Silico Biology, 2 (2002), 393-406. [3] L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER3 Optimal Control Software: Theory and User Manual. Version 3 The Department of Mathematics, The University of Western Australia, Nedlands, Western Australia, 2004. [4] T. A. Jones, Mathematical Modelling Of Cancer Growth Ph. D thesis, Curtin University in Bentley, Western Australia, 2014. [5] R. C. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929.  doi: 10.1016/j.automatica.2008.04.011. [6] R. B. Martin, Optimal control drug scheduling of cancer chemotherapy, Automatica, 28 (1992), 1113-1123.  doi: 10.1016/0005-1098(92)90054-J. [7] R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy World Scientific Publishing Company Pte. Ltd, Singapore, 1994. doi: 10.1142/2048. [8] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems Longman Scientific & Technical, Essex, England, 1991. [9] J. J. Tyson and B. Novak, Regulation of the eukaryotic cell cycle: Molecular antagonism, hysteresis, and irreversible transitions, JTB, 210 (2001), 249-263.  doi: 10.1006/jtbi.2001.2293. [10] World Health Organization, Media Centre -Cancer Electronically on World Wide Web, 2012. Available from: http://www.who.int/mediacentre/factsheets/fs297/en/. [11] R. A. Weinberg, The Biology of Cancer Garland Science, New York, 2007.

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##### References:
 [1] T. Alarcón, H. M. Byrne and P. K. Maini, A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells, JTB, 229 (2004), 397-411. [2] S. Dormann and A. Deutsch, Modeling of self-organized avascular tumor growth with a hybrid cellular automaton, In Silico Biology, 2 (2002), 393-406. [3] L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER3 Optimal Control Software: Theory and User Manual. Version 3 The Department of Mathematics, The University of Western Australia, Nedlands, Western Australia, 2004. [4] T. A. Jones, Mathematical Modelling Of Cancer Growth Ph. D thesis, Curtin University in Bentley, Western Australia, 2014. [5] R. C. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929.  doi: 10.1016/j.automatica.2008.04.011. [6] R. B. Martin, Optimal control drug scheduling of cancer chemotherapy, Automatica, 28 (1992), 1113-1123.  doi: 10.1016/0005-1098(92)90054-J. [7] R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy World Scientific Publishing Company Pte. Ltd, Singapore, 1994. doi: 10.1142/2048. [8] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems Longman Scientific & Technical, Essex, England, 1991. [9] J. J. Tyson and B. Novak, Regulation of the eukaryotic cell cycle: Molecular antagonism, hysteresis, and irreversible transitions, JTB, 210 (2001), 249-263.  doi: 10.1006/jtbi.2001.2293. [10] World Health Organization, Media Centre -Cancer Electronically on World Wide Web, 2012. Available from: http://www.who.int/mediacentre/factsheets/fs297/en/. [11] R. A. Weinberg, The Biology of Cancer Garland Science, New York, 2007.
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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