# American Institute of Mathematical Sciences

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). Google Scholar [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. Google Scholar [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. Google Scholar [4] I. D. Bassukas, Comparative Gompertzian analysis of alterations of tumor growth patterns,, Cancer Research, 54 (1994), 4385. Google Scholar [5] M. Bodnar and U. Foryś, Three types of simple DDEs describing tumour growth,, J. Biol. Sys., 15 (2007), 453. doi: 10.1142/S0218339007002313. Google Scholar [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. Google Scholar [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. Google Scholar [8] I. N. Bronshtein, K. A. Semendyayev, G. Musiol and H. Muehlig, "Handbook of Mathematics,", 5$^{th}$ edition, (2007). Google Scholar [9] L. Preziosi, "Cancer Modeling and Simulation,", Chapman & Hall, (2003). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] J. Folkman, Tumor angiogenesis: Therapeutic implications,, N. Engl. J. Med., 285 (1971), 1182. doi: 10.1056/NEJM197111182852108. Google Scholar [16] J. Folkman, Agiogenesis in cancer, vascular, rheumatoid and other disease,, Nat. Med., 1 (1995), 27. doi: 10.1038/nm0195-27. Google Scholar [17] J. Folkman, Angiogenesis,, Ann. Rev. Med., 57 (2006), 1. doi: 10.1146/annurev.med.57.121304.131306. Google Scholar [18] U. Foryś and A. Marciniak-Czochra, Logistic equation in tumour growth modelling,, Int. J. Appl. Math. Comp. Sci., 13 (2003), 317. Google Scholar [19] S. A. Frank, "Dynamics of Cancer - Incidence, Inheritance, and Evolution,", Princeton University Press, (2007). Google Scholar [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. Google Scholar [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. Google Scholar [22] R. K. Jain, Taming vessels to treat cancer,, Scientific American, 298 (2008), 56. doi: 10.1038/scientificamerican0108-56. Google Scholar [23] R. K. Jain, Normalization of tumour vasculature: An emerging concept in antiangiogenic therapy,, Science, 307 (2005), 58. doi: 10.1126/science.1104819. Google Scholar [24] A. K. Laird, Dynamics of tumour growth,, Br. J. Cancer, 18 (1964), 490. doi: 10.1038/bjc.1964.55. Google Scholar [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. Google Scholar [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. Google Scholar [27] U. Ledzewicz and H. Schättler, Optimal control for a system modeling tumor anti-angiogenesis,, in, (2005), 147. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [32] J. D. Murray, "Mathematical Biology. An Introduction,", Springer Verlag, (2002). Google Scholar [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. Google Scholar [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. Google Scholar [35] J. Poleszczuk, Tumor development model under angiogenic signaling with dependence on vessel impairment,, in, (2008), 104. Google Scholar [36] A. Świerniak, Comparison of six models of antiangiogenic therapy,, Appl. Math. (Warsaw), 36 (2009), 333. Google Scholar [37] A. Świerniak, G. Gala, A. Gandolfi and A. d'Onofrio, Optimization of anti-angiogenic therapy as optimal control problem,, in, (2006), 56. Google Scholar [38] A. Świerniak, A. d'Onofrio and A. Gandolfi, Control problems related to tumor angiogenesis,, in, (2006), 677. Google Scholar [39] P. E. Thorpe, Vascular targeting agents as cancer therapeutics,, Clin Cancer Res., 10 (2004), 415. doi: 10.1158/1078-0432.CCR-0642-03. Google Scholar [40] T. E. Wheldon, "Mathematical Models in Cancer Research,", Hilger Publishing, (1998). Google Scholar [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. Google Scholar

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). Google Scholar [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. Google Scholar [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. Google Scholar [4] I. D. Bassukas, Comparative Gompertzian analysis of alterations of tumor growth patterns,, Cancer Research, 54 (1994), 4385. Google Scholar [5] M. Bodnar and U. Foryś, Three types of simple DDEs describing tumour growth,, J. Biol. Sys., 15 (2007), 453. doi: 10.1142/S0218339007002313. Google Scholar [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. Google Scholar [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. Google Scholar [8] I. N. Bronshtein, K. A. Semendyayev, G. Musiol and H. Muehlig, "Handbook of Mathematics,", 5$^{th}$ edition, (2007). Google Scholar [9] L. Preziosi, "Cancer Modeling and Simulation,", Chapman & Hall, (2003). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] J. Folkman, Tumor angiogenesis: Therapeutic implications,, N. Engl. J. Med., 285 (1971), 1182. doi: 10.1056/NEJM197111182852108. Google Scholar [16] J. Folkman, Agiogenesis in cancer, vascular, rheumatoid and other disease,, Nat. Med., 1 (1995), 27. doi: 10.1038/nm0195-27. Google Scholar [17] J. Folkman, Angiogenesis,, Ann. Rev. Med., 57 (2006), 1. doi: 10.1146/annurev.med.57.121304.131306. Google Scholar [18] U. Foryś and A. Marciniak-Czochra, Logistic equation in tumour growth modelling,, Int. J. Appl. Math. Comp. Sci., 13 (2003), 317. Google Scholar [19] S. A. Frank, "Dynamics of Cancer - Incidence, Inheritance, and Evolution,", Princeton University Press, (2007). Google Scholar [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. Google Scholar [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. Google Scholar [22] R. K. Jain, Taming vessels to treat cancer,, Scientific American, 298 (2008), 56. doi: 10.1038/scientificamerican0108-56. Google Scholar [23] R. K. Jain, Normalization of tumour vasculature: An emerging concept in antiangiogenic therapy,, Science, 307 (2005), 58. doi: 10.1126/science.1104819. Google Scholar [24] A. K. Laird, Dynamics of tumour growth,, Br. J. Cancer, 18 (1964), 490. doi: 10.1038/bjc.1964.55. Google Scholar [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. Google Scholar [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. Google Scholar [27] U. Ledzewicz and H. Schättler, Optimal control for a system modeling tumor anti-angiogenesis,, in, (2005), 147. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [32] J. D. Murray, "Mathematical Biology. An Introduction,", Springer Verlag, (2002). Google Scholar [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. Google Scholar [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. Google Scholar [35] J. Poleszczuk, Tumor development model under angiogenic signaling with dependence on vessel impairment,, in, (2008), 104. Google Scholar [36] A. Świerniak, Comparison of six models of antiangiogenic therapy,, Appl. Math. (Warsaw), 36 (2009), 333. Google Scholar [37] A. Świerniak, G. Gala, A. Gandolfi and A. d'Onofrio, Optimization of anti-angiogenic therapy as optimal control problem,, in, (2006), 56. Google Scholar [38] A. Świerniak, A. d'Onofrio and A. Gandolfi, Control problems related to tumor angiogenesis,, in, (2006), 677. Google Scholar [39] P. E. Thorpe, Vascular targeting agents as cancer therapeutics,, Clin Cancer Res., 10 (2004), 415. doi: 10.1158/1078-0432.CCR-0642-03. Google Scholar [40] T. E. Wheldon, "Mathematical Models in Cancer Research,", Hilger Publishing, (1998). Google Scholar [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. Google Scholar
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