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Asymptotic properties of delayed matrix exponential functions via Lambert function
Sensitivity of combined chemo-and antiangiogenic therapy results in different models describing cancer growth
Systems Engineering Group, Silesian University of Technology, Akademicka 16, Gliwice, 44-100, Poland |
This paper is concerned with analysis of two anticancer therapy models focused on sensitivity of therapy outcome with respect to model structure and parameters. Realistic periodic therapies are considered, combining cytotoxic and antiangiogenic agents, defined on a fixed time horizon. Tumor size at the end of therapy and average tumor size calculated over therapy horizon are chosen to represent therapy outcome. Sensitivity analysis has been performed numerically, concentrating on model parameters and structure at one hand, and on treatment protocol parameters, on the other. The results show that sensitivity of the therapy outcome highly depends on the model structure, helping to discern a good model. Moreover, it is possible to use this analysis to find a good protocol in case of heterogeneous tumors.
References:
[1] |
K. D. Argyri, D. D. Dionysiou, F. D. Misichroni and G. S. Stamatakos, Numerical simulation of vascular tumour growth under antiangiogenic treatment: Addressing the paradigm of single-agent bevacizumab therapy with the use of experimental data Biology Direct, 11 (2016), p12.
doi: 10.1186/s13062-016-0114-9. |
[2] |
F. Billy, J. Clairambault and O. Fercoq, Optimisation of cancer drug treatments using cell population dynamics, Mathematical Methods and Models in Biomedicine (eds. Ledzewicz U., Schättler H., Friedman A., Kashdan E. ), (2013), 265-309.
doi: 10.1007/978-1-4614-4178-6_10. |
[3] | |
[4] |
B. C. Daniels, Y. J. Chen, J. P. Sethna, R. N. Gutenkunst and C. R. Myers,
Sloppiness, robustness, and evolvability in systems biology, Current Opinion in Biotechnology, 19 (2008), 389-395.
doi: 10.1016/j.copbio.2008.06.008. |
[5] |
A. D'Onofrio and A. Gandolfi,
A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Mathematical Medicine and Biology, 26 (2009), 63-95.
doi: 10.1093/imammb/dqn024. |
[6] |
A. d'Onofrio and A. Gandolfi,
Chemotherapy of vascularised tumours: Role of vessel density and the effect of vascular "pruning", Journal of Theoretical Biology, 264 (2010), 253-265.
doi: 10.1016/j.jtbi.2010.01.023. |
[7] |
A. D'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler,
On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222 (2009), 13-26.
doi: 10.1016/j.mbs.2009.08.004. |
[8] |
H. Enderling and M. A. Chaplain,
Mathematical modeling of tumor growth and treatment, Current Pharmaceutical Design, 20 (2014), 4934-4940.
doi: 10.2174/1381612819666131125150434. |
[9] |
J. Folkman and M. Klagsbrun,
Angiogenic factors, Science, 235 (1987), 442-447.
doi: 10.1126/science.2432664. |
[10] |
R. N. Gutenkunst, J. J. Watefall, F. P. Casey, K. S. Brown, C. R. Myers and J. P. Sethna,
Universally sloppy parameter sensitivities in systems biology models, PLOS Computational Biology, 3 (2007), 1871-1878.
doi: 10.1371/journal.pcbi.0030189. |
[11] |
P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky,
Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 19 (1999), 4770-4775.
|
[12] |
http://drugs.com, Available: 2016-09-30. |
[13] |
M. Komorowski, M. J. Costa, D. A. Rand and M. P. H. Stumpf,
Sensitivity, robustness, and identifiability in stochastic chemical kinetics models, PNAS, 108 (2011), 8645-8650.
doi: 10.1073/pnas.1015814108. |
[14] |
U. Ledzewicz, H. Maurer and H. Schättler, Minimizing Tumor Volume for a Mathematical Model of Anti-Angiogenesis with Linear Pharmacokinetics, Recent Advances in Optimization and its Applications in Engineering (eds. Diehl M., Glineur F., Jarlebring E., Michiels W., (2010), 267-276.
doi: 10.1007/978-3-642-12598-0_23. |
[15] |
U. Ledzewicz and H. Schättler,
Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
[16] |
B. K. Mannakee, A. P. Ragsdale, M. K. Transtrum and R. N. Gutenkunst,
Sloppiness and the geometry of parameter space, Uncertainty in Biology: A Computational Modeling Approach (eds. Geris L., Gomez-Cabrero D.), 17 (2015), 271-299.
doi: 10.1007/978-3-319-21296-8_11. |
[17] |
J. Poleszczuk, P. Hahnfeldt and H. Enderling, Therapeutic implications from sensitivity analysis of tumor angiogenesis models PLoS One, 10(2015), e0120007.
doi: 10.1371/journal.pone.0120007. |
[18] |
A. Saltelli, Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models, John
Wiley & Sons, 2004. |
[19] |
A. Saltelli, M. Ratto, S. Tarantola and F. Campolongo,
Sensitivity analysis practices: Strategies for model-based inference, Reliability Engineering and System Safety, (2006), 1109-1125.
|
[20] |
H. Schättler and U. Ledzewicz,
Optimal Control for Mathematical Models of Cancer Therapies -an Application of Geometric Methods, Interdisciplinary Applied Mathematics, 42. Springer, New York, 2015.
doi: 10.1007/978-1-4939-2972-6. |
[21] |
A. Swierniak, M. Kimmel and J. Smieja,
Mathematical modeling as a tool for planning anticancer therapy, European Journal of Pharmacology, 625 (2009), 108-121.
doi: 10.1016/j.ejphar.2009.08.041. |
[22] |
A. Swierniak, M. Kimmel, J. Smieja, K. Puszynski and K. Psiuk-Maksymowicz,
System Engineering Approach to Planning Anticancer Therapies, Springer, 2016.
doi: 10.1007/978-3-319-28095-0. |
[23] |
N. S. Vasudev and A. R. Reynolds,
Anti-angiogenic therapy for cancer: Current progress, unresolved questions and future directions, Angiogenesis, 17 (2014), 471-494.
doi: 10.1007/s10456-014-9420-y. |
[24] |
J. Welti, S. Loges, S. Dimmeler and P. Carmeliet,
Recent molecular discoveries in angiogenesis and antiangiogenic therapies in cancer, The Journal of Clinical Investigation, 8 (2013), 3190-3200.
doi: 10.1172/JCI70212. |
[25] |
J. Zalevsky, A. K. Chamberlain, H. M. Horton, S. Karki, I. W. Leung, T. J. Sproule, G. A. Lazar, D. C. Roopenian and J. R. Desjarlais,
Enhanced antibody half-life improves in vivo activity, Nature Biotechnology, 28 (2010), 157-159.
doi: 10.1038/nbt.1601. |
show all references
References:
[1] |
K. D. Argyri, D. D. Dionysiou, F. D. Misichroni and G. S. Stamatakos, Numerical simulation of vascular tumour growth under antiangiogenic treatment: Addressing the paradigm of single-agent bevacizumab therapy with the use of experimental data Biology Direct, 11 (2016), p12.
doi: 10.1186/s13062-016-0114-9. |
[2] |
F. Billy, J. Clairambault and O. Fercoq, Optimisation of cancer drug treatments using cell population dynamics, Mathematical Methods and Models in Biomedicine (eds. Ledzewicz U., Schättler H., Friedman A., Kashdan E. ), (2013), 265-309.
doi: 10.1007/978-1-4614-4178-6_10. |
[3] | |
[4] |
B. C. Daniels, Y. J. Chen, J. P. Sethna, R. N. Gutenkunst and C. R. Myers,
Sloppiness, robustness, and evolvability in systems biology, Current Opinion in Biotechnology, 19 (2008), 389-395.
doi: 10.1016/j.copbio.2008.06.008. |
[5] |
A. D'Onofrio and A. Gandolfi,
A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Mathematical Medicine and Biology, 26 (2009), 63-95.
doi: 10.1093/imammb/dqn024. |
[6] |
A. d'Onofrio and A. Gandolfi,
Chemotherapy of vascularised tumours: Role of vessel density and the effect of vascular "pruning", Journal of Theoretical Biology, 264 (2010), 253-265.
doi: 10.1016/j.jtbi.2010.01.023. |
[7] |
A. D'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler,
On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222 (2009), 13-26.
doi: 10.1016/j.mbs.2009.08.004. |
[8] |
H. Enderling and M. A. Chaplain,
Mathematical modeling of tumor growth and treatment, Current Pharmaceutical Design, 20 (2014), 4934-4940.
doi: 10.2174/1381612819666131125150434. |
[9] |
J. Folkman and M. Klagsbrun,
Angiogenic factors, Science, 235 (1987), 442-447.
doi: 10.1126/science.2432664. |
[10] |
R. N. Gutenkunst, J. J. Watefall, F. P. Casey, K. S. Brown, C. R. Myers and J. P. Sethna,
Universally sloppy parameter sensitivities in systems biology models, PLOS Computational Biology, 3 (2007), 1871-1878.
doi: 10.1371/journal.pcbi.0030189. |
[11] |
P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky,
Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 19 (1999), 4770-4775.
|
[12] |
http://drugs.com, Available: 2016-09-30. |
[13] |
M. Komorowski, M. J. Costa, D. A. Rand and M. P. H. Stumpf,
Sensitivity, robustness, and identifiability in stochastic chemical kinetics models, PNAS, 108 (2011), 8645-8650.
doi: 10.1073/pnas.1015814108. |
[14] |
U. Ledzewicz, H. Maurer and H. Schättler, Minimizing Tumor Volume for a Mathematical Model of Anti-Angiogenesis with Linear Pharmacokinetics, Recent Advances in Optimization and its Applications in Engineering (eds. Diehl M., Glineur F., Jarlebring E., Michiels W., (2010), 267-276.
doi: 10.1007/978-3-642-12598-0_23. |
[15] |
U. Ledzewicz and H. Schättler,
Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
[16] |
B. K. Mannakee, A. P. Ragsdale, M. K. Transtrum and R. N. Gutenkunst,
Sloppiness and the geometry of parameter space, Uncertainty in Biology: A Computational Modeling Approach (eds. Geris L., Gomez-Cabrero D.), 17 (2015), 271-299.
doi: 10.1007/978-3-319-21296-8_11. |
[17] |
J. Poleszczuk, P. Hahnfeldt and H. Enderling, Therapeutic implications from sensitivity analysis of tumor angiogenesis models PLoS One, 10(2015), e0120007.
doi: 10.1371/journal.pone.0120007. |
[18] |
A. Saltelli, Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models, John
Wiley & Sons, 2004. |
[19] |
A. Saltelli, M. Ratto, S. Tarantola and F. Campolongo,
Sensitivity analysis practices: Strategies for model-based inference, Reliability Engineering and System Safety, (2006), 1109-1125.
|
[20] |
H. Schättler and U. Ledzewicz,
Optimal Control for Mathematical Models of Cancer Therapies -an Application of Geometric Methods, Interdisciplinary Applied Mathematics, 42. Springer, New York, 2015.
doi: 10.1007/978-1-4939-2972-6. |
[21] |
A. Swierniak, M. Kimmel and J. Smieja,
Mathematical modeling as a tool for planning anticancer therapy, European Journal of Pharmacology, 625 (2009), 108-121.
doi: 10.1016/j.ejphar.2009.08.041. |
[22] |
A. Swierniak, M. Kimmel, J. Smieja, K. Puszynski and K. Psiuk-Maksymowicz,
System Engineering Approach to Planning Anticancer Therapies, Springer, 2016.
doi: 10.1007/978-3-319-28095-0. |
[23] |
N. S. Vasudev and A. R. Reynolds,
Anti-angiogenic therapy for cancer: Current progress, unresolved questions and future directions, Angiogenesis, 17 (2014), 471-494.
doi: 10.1007/s10456-014-9420-y. |
[24] |
J. Welti, S. Loges, S. Dimmeler and P. Carmeliet,
Recent molecular discoveries in angiogenesis and antiangiogenic therapies in cancer, The Journal of Clinical Investigation, 8 (2013), 3190-3200.
doi: 10.1172/JCI70212. |
[25] |
J. Zalevsky, A. K. Chamberlain, H. M. Horton, S. Karki, I. W. Leung, T. J. Sproule, G. A. Lazar, D. C. Roopenian and J. R. Desjarlais,
Enhanced antibody half-life improves in vivo activity, Nature Biotechnology, 28 (2010), 157-159.
doi: 10.1038/nbt.1601. |











Par. | Values | Description | |
Model (2) | Model (3) | ||
Initial no. of cancer cells | |||
Initial no. of endothelial cells | |||
Tumor growth parameter | |||
Endothelial stimulation parameter | |||
Endothelial inhibition parameter | |||
0 | 0 | Natural mortality of endothelial cells | |
- | Cytostatic killing parameter (for cancer cells) | ||
- | Maximum value of cytostatic killing parameter (for cancer cells) - | ||
| Cytostatic killing parameter (for endothelial cells) | ||
| Anti-angiogenic killing parameter | ||
| - | Parametr used in the efficacy curve of the drug | |
| - | 2 |
Par. | Values | Description | |
Model (2) | Model (3) | ||
Initial no. of cancer cells | |||
Initial no. of endothelial cells | |||
Tumor growth parameter | |||
Endothelial stimulation parameter | |||
Endothelial inhibition parameter | |||
0 | 0 | Natural mortality of endothelial cells | |
- | Cytostatic killing parameter (for cancer cells) | ||
- | Maximum value of cytostatic killing parameter (for cancer cells) - | ||
| Cytostatic killing parameter (for endothelial cells) | ||
| Anti-angiogenic killing parameter | ||
| - | Parametr used in the efficacy curve of the drug | |
| - | 2 |
Parameter | ||||
Model (2) | 12[h] | 3.5[days] | ||
Model (3) | 12[h] | 3.5[days] | ||
Parameter | ||||
Model (2) | 12[h] | 3.5[days] | ||
Model (3) | 12[h] | 3.5[days] | ||
Case | (ⅰ) | (ⅱ) | ||||
Parameter | ||||||
Value |
Case | (ⅰ) | (ⅱ) | ||||
Parameter | ||||||
Value |
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