American Institute of Mathematical Sciences

Advanced
January  2018, 23(1): 145-160. doi: 10.3934/dcdsb.2018009

Sensitivity of combined chemo-and antiangiogenic therapy results in different models describing cancer growth

Marzena Dolbniak , Malgorzata Kardynska and  Jaroslaw Smieja ,

Systems Engineering Group, Silesian University of Technology, Akademicka 16, Gliwice, 44-100, Poland

* Corresponding author: Jaroslaw Smieja

Received  November 2016 Revised  June 2017 Published  January 2018

Fund Project: This work was supported by the NCN grant DEC-2013/11/B/ST7/01713.

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.

Keywords: Chemotherapy, antiangiogenic therapy, sensitivity.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Marzena Dolbniak, Malgorzata Kardynska, Jaroslaw Smieja. Sensitivity of combined chemo-and antiangiogenic therapy results in different models describing cancer growth. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 145-160. doi: 10.3934/dcdsb.2018009
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]

J. J. Cruz, Feedback Systems, McGraw-Hill, 1972.

[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]

J. J. Cruz, Feedback Systems, McGraw-Hill, 1972.

[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.

Figure 1.  Parameter sensitivity ranking calculated with first-order sensitivity functions for the model (2). Sensitivity indices for parameters were normalized to the maximum value
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Parameter sensitivity rankings calculated with first-order sensitivity functions for different treatment protocols and the model (2): chemotherapy administered every (a) 2 days, (b) 4 days, (c) 10 days; antiangiogenic agents administered every (a) 10 days, (b) 13 days, (c) 20 days. Sensitivity indices for parameters were normalized to the maximum value to facilitate comparison of the rankings
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The heatmap representing sensitivity of the treatment outcome, defined as the number of cancer cells at the end of treatment (right) and two sample solutions that illustrate its meaning (left)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Evaluation of various treatment protocols for the models (a, b) (2) and (c, d) (3); taking into account: low drugs toxicity (a, c) and high drugs toxicity (b, d). Axes are labeled by the number of days between consecutive doses of respective drugs
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  An example of the model (2) response for treatment protocol selected on the basis of J
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Changes in therapy outcome caused by different administration times of the same total doses for the models (a, b) (2) and (c, d) (3). Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Changes in therapy outcome for model (2) due to changes in parameter $\beta$: (a, b) increased by 50%; (c, d) decreased by 50%. Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Changes in therapy efficiency caused by variability in parameters for model (2). Maximum change is presented. (a, b) $\beta$, (c, d) $\gamma$, (e, f) $\lambda$. Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Changes in therapy efficiency caused by variability in parameters for model (2). Mean change is presented. (a, b) $\beta$, (c, d) $\gamma$, (e, f) $\lambda$. Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  Changes in therapy efficiency caused by variability in parameters for model (3). Max change is presented. (a, b) $\beta$, (c, d) $\gamma$, (e, f) $\lambda$. Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  Changes in therapy efficiency caused by variability in parameters for model (3). Mean change is presented. (a, b) $\beta$, (c, d) $\gamma$, (e, f) $\lambda$. Left and right columns represent indices given by the tumor volume at the end of treatment and mean tumor volume during the treatment, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Nominal values of parameters
Par.ValuesDescription
Model (2)Model (3)
$N_0$$10^6$$10^6$Initial no. of cancer cells $[mm^3]$
$K_0$$N_0 \cdot 10 $$N_0 \cdot 10 $Initial no. of endothelial cells $[mm^3]$
$\beta$$1.92 \cdot 10^{-1}$$1.92 \cdot 10^{-1}$Tumor growth parameter $[day^{-1}]$
$\gamma$$1.755 \cdot 10^{1}$$1.755 \cdot 10^{1}$Endothelial stimulation parameter $[day^{-1}]$
$\lambda$$8.73 \cdot 10^{-5}$$8.73 \cdot 10^{-5}$Endothelial inhibition parameter $[day^{-1} mm^{-\frac{2}{3}}]$
$\mu$00Natural mortality of endothelial cells $[day^{-1}]$
$\psi$$1.17 \cdot 10^{-2}$-Cytostatic killing parameter (for cancer cells) $[mg^{-1} m^2]$
$\psi * $-$1.17 \cdot 10^{-2}$Maximum value of cytostatic killing parameter (for cancer cells) -$[mg^{-1} m^2]$
$\eta$$1.17 \cdot 10^{-4}$$1.17 \cdot 10^{-4}$Cytostatic killing parameter (for endothelial cells) $[mg^{-1} m^2]$
$\xi$$1.75 \cdot 10^{-2}$$5.25 \cdot 10^{-2}$Anti-angiogenic killing parameter $[mg^{-1} kg]$
$\sigma$-$3.5\cdot 10^{-1}$Parametr used in the efficacy curve of the drug
$\rho_{opt}$-2$\frac{K}{N}$ ratio, for which the most effective functionality of vasculature is observed
Table Options

Download as excel

Par.ValuesDescription
Model (2)Model (3)
$N_0$$10^6$$10^6$Initial no. of cancer cells $[mm^3]$
$K_0$$N_0 \cdot 10 $$N_0 \cdot 10 $Initial no. of endothelial cells $[mm^3]$
$\beta$$1.92 \cdot 10^{-1}$$1.92 \cdot 10^{-1}$Tumor growth parameter $[day^{-1}]$
$\gamma$$1.755 \cdot 10^{1}$$1.755 \cdot 10^{1}$Endothelial stimulation parameter $[day^{-1}]$
$\lambda$$8.73 \cdot 10^{-5}$$8.73 \cdot 10^{-5}$Endothelial inhibition parameter $[day^{-1} mm^{-\frac{2}{3}}]$
$\mu$00Natural mortality of endothelial cells $[day^{-1}]$
$\psi$$1.17 \cdot 10^{-2}$-Cytostatic killing parameter (for cancer cells) $[mg^{-1} m^2]$
$\psi * $-$1.17 \cdot 10^{-2}$Maximum value of cytostatic killing parameter (for cancer cells) -$[mg^{-1} m^2]$
$\eta$$1.17 \cdot 10^{-4}$$1.17 \cdot 10^{-4}$Cytostatic killing parameter (for endothelial cells) $[mg^{-1} m^2]$
$\xi$$1.75 \cdot 10^{-2}$$5.25 \cdot 10^{-2}$Anti-angiogenic killing parameter $[mg^{-1} kg]$
$\sigma$-$3.5\cdot 10^{-1}$Parametr used in the efficacy curve of the drug
$\rho_{opt}$-2$\frac{K}{N}$ ratio, for which the most effective functionality of vasculature is observed
Table 2.  Pharmacokinetics parameters
Parameter$k_u$$k_v$$T_{u1/2}$$T_{v1/2}$
Model (2)$0.0117$$0.0175$12[h]3.5[days]
Model (3)$0.0117$$0.0525$12[h]3.5[days]
$u$ -chemotherapy
$v$ -antiangiogenic therapy
Table Options

Download as excel

Parameter$k_u$$k_v$$T_{u1/2}$$T_{v1/2}$
Model (2)$0.0117$$0.0175$12[h]3.5[days]
Model (3)$0.0117$$0.0525$12[h]3.5[days]
$u$ -chemotherapy
$v$ -antiangiogenic therapy
Table 3.  Performance index parameters for models (2) and (3)
Case(ⅰ)(ⅱ)
Parameter$r_1$$r_2$$r_3$$r_1$$r_2$$r_3$
Value$1$$0.1 \cdot 10^4$$0.05 \cdot 10^4$$1$$0.5 \cdot 10^5$$0.25 \cdot 10^5$
Table Options

Download as excel

Case(ⅰ)(ⅱ)
Parameter$r_1$$r_2$$r_3$$r_1$$r_2$$r_3$
Value$1$$0.1 \cdot 10^4$$0.05 \cdot 10^4$$1$$0.5 \cdot 10^5$$0.25 \cdot 10^5$
[1]

Tania Biswas, Elisabetta Rocca. Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2455-2469. doi: 10.3934/dcdsb.2021140
[2]

Adam Glick, Antonio Mastroberardino. Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5281-5304. doi: 10.3934/dcdsb.2020343
[3]

Cristian Morales-Rodrigo. A therapy inactivating the tumor angiogenic factors. Mathematical Biosciences & Engineering, 2013, 10 (1) : 185-198. doi: 10.3934/mbe.2013.10.185
[4]

Ismail Abdulrashid, Abdallah A. M. Alsammani, Xiaoying Han. Stability analysis of a chemotherapy model with delays. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 989-1005. doi: 10.3934/dcdsb.2019002
[5]

Ismail Abdulrashid, Xiaoying Han. A mathematical model of chemotherapy with variable infusion. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1875-1890. doi: 10.3934/cpaa.2020082
[6]

Jan Poleszczuk, Marek Bodnar, Urszula Foryś. New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model. Mathematical Biosciences & Engineering, 2011, 8 (2) : 591-603. doi: 10.3934/mbe.2011.8.591
[7]

Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565
[8]

Manuel Delgado, Cristian Morales-Rodrigo, Antonio Suárez. Anti-angiogenic therapy based on the binding to receptors. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3871-3894. doi: 10.3934/dcds.2012.32.3871
[9]

Piotr Bajger, Mariusz Bodzioch, Urszula Foryś. Singularity of controls in a simple model of acquired chemotherapy resistance. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2039-2052. doi: 10.3934/dcdsb.2019083
[10]

Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129
[11]

Urszula Ledzewicz, Heinz Schättler. Controlling a model for bone marrow dynamics in cancer chemotherapy. Mathematical Biosciences & Engineering, 2004, 1 (1) : 95-110. doi: 10.3934/mbe.2004.1.95
[12]

Craig Collins, K. Renee Fister, Bethany Key, Mary Williams. Blasting neuroblastoma using optimal control of chemotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 451-467. doi: 10.3934/mbe.2009.6.451
[13]

Urszula Ledzewicz, Behrooz Amini, Heinz Schättler. Dynamics and control of a mathematical model for metronomic chemotherapy. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1257-1275. doi: 10.3934/mbe.2015.12.1257
[14]

Urszula Ledzewicz, Heinz Schättler, Shuo Wang. On the role of tumor heterogeneity for optimal cancer chemotherapy. Networks and Heterogeneous Media, 2019, 14 (1) : 131-147. doi: 10.3934/nhm.2019007
[15]

Danthai Thongphiew, Vira Chankong, Fang-Fang Yin, Q. Jackie Wu. An on-line adaptive radiation therapy system for intensity modulated radiation therapy: An application of multi-objective optimization. Journal of Industrial and Management Optimization, 2008, 4 (3) : 453-475. doi: 10.3934/jimo.2008.4.453
[16]

Caglar S. Aksezer. On the sensitivity of desirability functions for multiresponse optimization. Journal of Industrial and Management Optimization, 2008, 4 (4) : 685-696. doi: 10.3934/jimo.2008.4.685
[17]

Tao Yu. Measurable sensitivity via Furstenberg families. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4543-4563. doi: 10.3934/dcds.2017194
[18]

Jianjun Paul Tian. Finite-time perturbations of dynamical systems and applications to tumor therapy. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 469-479. doi: 10.3934/dcdsb.2009.12.469
[19]

Urszula Ledzewicz, Helen Moore. Optimal control applied to a generalized Michaelis-Menten model of CML therapy. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 331-346. doi: 10.3934/dcdsb.2018022
[20]

Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6825-6842. doi: 10.3934/dcds.2019233

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (84)
  • HTML views (160)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy