2011, 8(2): 591-603. doi: 10.3934/mbe.2011.8.591

New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model

1. 

University of Warsaw, Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw, Poland

Received  March 2010 Revised  August 2010 Published  April 2011

In the paper we propose a new methodology in modeling of antiangiogenic treatment on the basis of well recognized model formulated by Hahnfeldt et al. in 1999. On the basis of the Hahnfeldt et al. model, with the usage of the optimal control theory, some protocols of antiangiogenic treatment were proposed. However, in our opinion the formulation of that model is valid only for the antivascular treatment, that is treatment that is focused on destroying endothelial cells. Therefore, we propose a modification of the original model which is valid in the case of the antiangiogenic treatment, that is treatment which is focused on blocking angiogenic signaling. We analyze basic mathematical properties of the proposed model and present some numerical simulations.
Citation: 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
References:
[1]

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, "Molecular Biology of Cell,", Garland Publishing, (2007).

[2]

Z. Agur, L. Arakelyan, P. Daugulis and Y. Ginosar, Hopf point analysis for angiogenesis models,, Discrete & Cont. Dyn. Sys. B, 4 (2004), 29. doi: 10.3934/dcdsb.2004.4.29.

[3]

L. Arakelyan, V. Vainstein and Z. Agur, A computer algorithm describing the process of vessel formation and maturation, and its use for predicting the effects of anti-angiogenic and anti-maturation therapy on vascular tumor growth,, Angiogenesis, 5 (2002), 203. doi: 10.1023/A:1023841921971.

[4]

I. D. Bassukas, Comparative Gompertzian analysis of alterations of tumor growth patterns,, Cancer Research, 54 (1994), 4385.

[5]

M. Bodnar and U. Foryś, Three types of simple DDEs describing tumour growth,, J. Biol. Sys., 15 (2007), 453. doi: 10.1142/S0218339007002313.

[6]

M. Bodnar and U. Foryś, Angiogenesis model with carrying capacity depending on vessel density,, J. Biol. Sys., 17 (2009), 1. doi: 10.1142/S0218339009002739.

[7]

M. Bodnar and U. Foryś, Influence of time delays on the Hahnfeldt et al. angiogenesis model dynamics,, Appl. Math. (Warsaw), 36 (2009), 251. doi: 10.4064/am36-3-1.

[8]

I. N. Bronshtein, K. A. Semendyayev, G. Musiol and H. Muehlig, "Handbook of Mathematics,", 5$^{th}$ edition, (2007).

[9]

L. Preziosi, "Cancer Modeling and Simulation,", Chapman & Hall, (2003).

[10]

A. d'Onofrio and A. Gandolfi, Tumor eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999),, Math. Biosci., 191 (2004), 159. doi: 10.1016/j.mbs.2004.06.003.

[11]

A. d'Onofrio and A. Gandolfi, The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation,, Appl. Math. Comput., 181 (2006), 1155. doi: 10.1016/j.amc.2006.01.061.

[12]

A. d'Onofrio and A. Gandolfi, A family of models of angiogenesis and antiangiogensis anticancer therapy,, Math. Med. Biol., 26 (2009), 63. doi: 10.1093/imammb/dqn024.

[13]

A. d'Onofrio, A. Gandolfi and A. Rocca, The dynamics of tumour-vasculature interaction suggests low-dose, time-dense anti-angiogenic schedulings,, Cell Prolif., 42 (2009), 317. doi: 10.1111/j.1365-2184.2009.00595.x.

[14]

A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors,, Bull. Math. Biol, 65 (2003), 407. doi: 10.1016/S0092-8240(03)00006-5.

[15]

J. Folkman, Tumor angiogenesis: Therapeutic implications,, N. Engl. J. Med., 285 (1971), 1182. doi: 10.1056/NEJM197111182852108.

[16]

J. Folkman, Agiogenesis in cancer, vascular, rheumatoid and other disease,, Nat. Med., 1 (1995), 27. doi: 10.1038/nm0195-27.

[17]

J. Folkman, Angiogenesis,, Ann. Rev. Med., 57 (2006), 1. doi: 10.1146/annurev.med.57.121304.131306.

[18]

U. Foryś and A. Marciniak-Czochra, Logistic equation in tumour growth modelling,, Int. J. Appl. Math. Comp. Sci., 13 (2003), 317.

[19]

S. A. Frank, "Dynamics of Cancer - Incidence, Inheritance, and Evolution,", Princeton University Press, (2007).

[20]

B. Gompertz, On the nature of the function expressive of the law of human mortality, and a new mode of determining the value of life contingencies,, Phil. Trans. Roy. Soc., 115 (1825), 513. doi: 10.1098/rstl.1825.0026.

[21]

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 Res., 59 (1999), 4770.

[22]

R. K. Jain, Taming vessels to treat cancer,, Scientific American, 298 (2008), 56. doi: 10.1038/scientificamerican0108-56.

[23]

R. K. Jain, Normalization of tumour vasculature: An emerging concept in antiangiogenic therapy,, Science, 307 (2005), 58. doi: 10.1126/science.1104819.

[24]

A. K. Laird, Dynamics of tumour growth,, Br. J. Cancer, 18 (1964), 490. doi: 10.1038/bjc.1964.55.

[25]

A. K. Laird, Dynamics of tumour growth: Comparison of growth rates and extrapolation of growth curve to one cell,, Br. J. Cancer, 19 (1965), 278. doi: 10.1038/bjc.1965.32.

[26]

M. O. Leach, K. M. Brindle, J. L. Evelhoch, J. R. Griffiths, M. R. Horsman, A. Jackson, G. C. Jayson, I. R. Judson, M. V. Knopp, R. J. Maxwell, D. McIntyre, A. R. Padhani, P. Price, R. Rathbone, G. J. Rustin, P. S. Tofts, G. M. Tozer, W. Vennart, J. C. Waterton, S. R. Williams and P. Workman, The assessment of antiangiogenic and antivascular therapies in early-stage clinical trials using magnetic resonance imaging: Issues and recommendations,, Br. J. Cancer, 92 (2005), 1599. doi: 10.1038/sj.bjc.6602550.

[27]

U. Ledzewicz and H. Schättler, Optimal control for a system modeling tumor anti-angiogenesis,, in, (2005), 147.

[28]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. Control Optim., 46 (2007), 1052. doi: 10.1137/060665294.

[29]

U. Ledzewicz and H. Schättler, Analysis of a mathematical model for tumor anti-angiogenesis,, Optimal Control Appl. Methods, 29 (2008), 41. doi: 10.1002/oca.814.

[30]

U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models for tumor anti-angiogenesis,, J. Theor. Biol., 252 (2008), 295. doi: 10.1016/j.jtbi.2008.02.014.

[31]

N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis,, J. Math. Biol., 49 (2004), 111. doi: 10.1007/s00285-003-0262-2.

[32]

J. D. Murray, "Mathematical Biology. An Introduction,", Springer Verlag, (2002).

[33]

M. S. O'Reilly, L. Holmgren, Y. Shing, C. Chen, R. A. Rosenthal, M. Moses, W. S. Lane, Y. Cao, E. H. Sage and J. Folkman, Agiostatin: A novel angiogenesis inhibitor that mediates the suppression of metastases by a Lewis lung carcinoma,, Cell, 79 (1994), 315. doi: 10.1016/0092-8674(94)90200-3.

[34]

M. S. O'Reilly, T. Boehm, Y. Shing, N. Fukai, G. Vasios, W. S. Lane, E. Flynn, J. R. Birkhead, B. R. Olsen and J. Folkman, Endostatin: An endogenous inhibitor of angiogenesis and tumor growth,, Cell, 88 (1997), 277. doi: 10.1016/S0092-8674(00)81848-6.

[35]

J. Poleszczuk, Tumor development model under angiogenic signaling with dependence on vessel impairment,, in, (2008), 104.

[36]

A. Świerniak, Comparison of six models of antiangiogenic therapy,, Appl. Math. (Warsaw), 36 (2009), 333.

[37]

A. Świerniak, G. Gala, A. Gandolfi and A. d'Onofrio, Optimization of anti-angiogenic therapy as optimal control problem,, in, (2006), 56.

[38]

A. Świerniak, A. d'Onofrio and A. Gandolfi, Control problems related to tumor angiogenesis,, in, (2006), 677.

[39]

P. E. Thorpe, Vascular targeting agents as cancer therapeutics,, Clin Cancer Res., 10 (2004), 415. doi: 10.1158/1078-0432.CCR-0642-03.

[40]

T. E. Wheldon, "Mathematical Models in Cancer Research,", Hilger Publishing, (1998).

[41]

J. C. Yang, L. Haworth, R. M. Sherry, P. Hwu, D. J. Schwartzentruber, S. L. Topalian, S. M. Steinberg, H. X. Chen and Steven A. Rosenberg, A randomized trial of bevacizumab, an anti-vascular endothelial growth factor antibody, for metastatic renal cancer,, N. Engl. J. Med., 349 (2003), 427. doi: 10.1056/NEJMoa021491.

show all references

References:
[1]

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, "Molecular Biology of Cell,", Garland Publishing, (2007).

[2]

Z. Agur, L. Arakelyan, P. Daugulis and Y. Ginosar, Hopf point analysis for angiogenesis models,, Discrete & Cont. Dyn. Sys. B, 4 (2004), 29. doi: 10.3934/dcdsb.2004.4.29.

[3]

L. Arakelyan, V. Vainstein and Z. Agur, A computer algorithm describing the process of vessel formation and maturation, and its use for predicting the effects of anti-angiogenic and anti-maturation therapy on vascular tumor growth,, Angiogenesis, 5 (2002), 203. doi: 10.1023/A:1023841921971.

[4]

I. D. Bassukas, Comparative Gompertzian analysis of alterations of tumor growth patterns,, Cancer Research, 54 (1994), 4385.

[5]

M. Bodnar and U. Foryś, Three types of simple DDEs describing tumour growth,, J. Biol. Sys., 15 (2007), 453. doi: 10.1142/S0218339007002313.

[6]

M. Bodnar and U. Foryś, Angiogenesis model with carrying capacity depending on vessel density,, J. Biol. Sys., 17 (2009), 1. doi: 10.1142/S0218339009002739.

[7]

M. Bodnar and U. Foryś, Influence of time delays on the Hahnfeldt et al. angiogenesis model dynamics,, Appl. Math. (Warsaw), 36 (2009), 251. doi: 10.4064/am36-3-1.

[8]

I. N. Bronshtein, K. A. Semendyayev, G. Musiol and H. Muehlig, "Handbook of Mathematics,", 5$^{th}$ edition, (2007).

[9]

L. Preziosi, "Cancer Modeling and Simulation,", Chapman & Hall, (2003).

[10]

A. d'Onofrio and A. Gandolfi, Tumor eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999),, Math. Biosci., 191 (2004), 159. doi: 10.1016/j.mbs.2004.06.003.

[11]

A. d'Onofrio and A. Gandolfi, The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation,, Appl. Math. Comput., 181 (2006), 1155. doi: 10.1016/j.amc.2006.01.061.

[12]

A. d'Onofrio and A. Gandolfi, A family of models of angiogenesis and antiangiogensis anticancer therapy,, Math. Med. Biol., 26 (2009), 63. doi: 10.1093/imammb/dqn024.

[13]

A. d'Onofrio, A. Gandolfi and A. Rocca, The dynamics of tumour-vasculature interaction suggests low-dose, time-dense anti-angiogenic schedulings,, Cell Prolif., 42 (2009), 317. doi: 10.1111/j.1365-2184.2009.00595.x.

[14]

A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors,, Bull. Math. Biol, 65 (2003), 407. doi: 10.1016/S0092-8240(03)00006-5.

[15]

J. Folkman, Tumor angiogenesis: Therapeutic implications,, N. Engl. J. Med., 285 (1971), 1182. doi: 10.1056/NEJM197111182852108.

[16]

J. Folkman, Agiogenesis in cancer, vascular, rheumatoid and other disease,, Nat. Med., 1 (1995), 27. doi: 10.1038/nm0195-27.

[17]

J. Folkman, Angiogenesis,, Ann. Rev. Med., 57 (2006), 1. doi: 10.1146/annurev.med.57.121304.131306.

[18]

U. Foryś and A. Marciniak-Czochra, Logistic equation in tumour growth modelling,, Int. J. Appl. Math. Comp. Sci., 13 (2003), 317.

[19]

S. A. Frank, "Dynamics of Cancer - Incidence, Inheritance, and Evolution,", Princeton University Press, (2007).

[20]

B. Gompertz, On the nature of the function expressive of the law of human mortality, and a new mode of determining the value of life contingencies,, Phil. Trans. Roy. Soc., 115 (1825), 513. doi: 10.1098/rstl.1825.0026.

[21]

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 Res., 59 (1999), 4770.

[22]

R. K. Jain, Taming vessels to treat cancer,, Scientific American, 298 (2008), 56. doi: 10.1038/scientificamerican0108-56.

[23]

R. K. Jain, Normalization of tumour vasculature: An emerging concept in antiangiogenic therapy,, Science, 307 (2005), 58. doi: 10.1126/science.1104819.

[24]

A. K. Laird, Dynamics of tumour growth,, Br. J. Cancer, 18 (1964), 490. doi: 10.1038/bjc.1964.55.

[25]

A. K. Laird, Dynamics of tumour growth: Comparison of growth rates and extrapolation of growth curve to one cell,, Br. J. Cancer, 19 (1965), 278. doi: 10.1038/bjc.1965.32.

[26]

M. O. Leach, K. M. Brindle, J. L. Evelhoch, J. R. Griffiths, M. R. Horsman, A. Jackson, G. C. Jayson, I. R. Judson, M. V. Knopp, R. J. Maxwell, D. McIntyre, A. R. Padhani, P. Price, R. Rathbone, G. J. Rustin, P. S. Tofts, G. M. Tozer, W. Vennart, J. C. Waterton, S. R. Williams and P. Workman, The assessment of antiangiogenic and antivascular therapies in early-stage clinical trials using magnetic resonance imaging: Issues and recommendations,, Br. J. Cancer, 92 (2005), 1599. doi: 10.1038/sj.bjc.6602550.

[27]

U. Ledzewicz and H. Schättler, Optimal control for a system modeling tumor anti-angiogenesis,, in, (2005), 147.

[28]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. Control Optim., 46 (2007), 1052. doi: 10.1137/060665294.

[29]

U. Ledzewicz and H. Schättler, Analysis of a mathematical model for tumor anti-angiogenesis,, Optimal Control Appl. Methods, 29 (2008), 41. doi: 10.1002/oca.814.

[30]

U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models for tumor anti-angiogenesis,, J. Theor. Biol., 252 (2008), 295. doi: 10.1016/j.jtbi.2008.02.014.

[31]

N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis,, J. Math. Biol., 49 (2004), 111. doi: 10.1007/s00285-003-0262-2.

[32]

J. D. Murray, "Mathematical Biology. An Introduction,", Springer Verlag, (2002).

[33]

M. S. O'Reilly, L. Holmgren, Y. Shing, C. Chen, R. A. Rosenthal, M. Moses, W. S. Lane, Y. Cao, E. H. Sage and J. Folkman, Agiostatin: A novel angiogenesis inhibitor that mediates the suppression of metastases by a Lewis lung carcinoma,, Cell, 79 (1994), 315. doi: 10.1016/0092-8674(94)90200-3.

[34]

M. S. O'Reilly, T. Boehm, Y. Shing, N. Fukai, G. Vasios, W. S. Lane, E. Flynn, J. R. Birkhead, B. R. Olsen and J. Folkman, Endostatin: An endogenous inhibitor of angiogenesis and tumor growth,, Cell, 88 (1997), 277. doi: 10.1016/S0092-8674(00)81848-6.

[35]

J. Poleszczuk, Tumor development model under angiogenic signaling with dependence on vessel impairment,, in, (2008), 104.

[36]

A. Świerniak, Comparison of six models of antiangiogenic therapy,, Appl. Math. (Warsaw), 36 (2009), 333.

[37]

A. Świerniak, G. Gala, A. Gandolfi and A. d'Onofrio, Optimization of anti-angiogenic therapy as optimal control problem,, in, (2006), 56.

[38]

A. Świerniak, A. d'Onofrio and A. Gandolfi, Control problems related to tumor angiogenesis,, in, (2006), 677.

[39]

P. E. Thorpe, Vascular targeting agents as cancer therapeutics,, Clin Cancer Res., 10 (2004), 415. doi: 10.1158/1078-0432.CCR-0642-03.

[40]

T. E. Wheldon, "Mathematical Models in Cancer Research,", Hilger Publishing, (1998).

[41]

J. C. Yang, L. Haworth, R. M. Sherry, P. Hwu, D. J. Schwartzentruber, S. L. Topalian, S. M. Steinberg, H. X. Chen and Steven A. Rosenberg, A randomized trial of bevacizumab, an anti-vascular endothelial growth factor antibody, for metastatic renal cancer,, N. Engl. J. Med., 349 (2003), 427. doi: 10.1056/NEJMoa021491.

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