# American Institute of Mathematical Sciences

August  2018, 15(4): 827-839. doi: 10.3934/mbe.2018037

## Closed-loop control of tumor growth by means of anti-angiogenic administration

 1 Università Campus Bio-medico di Roma, Roma, Via Álvaro del Portillo 21, 00128, Italy 2 Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, Consiglio Nazionale delle Ricerche, Roma, Via dei Taurini 19, 00185, Italy 3 Dipartimento di Ingegneria e scienze dell'informazione e matematica, Università dell'Aquila, L'Aquila, Via Vetoio, 67100, Italy 4 Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, Consiglio Nazionale delle Ricerche, Roma, Via dei Taurini 19, 00185, Italy

* Corresponding author: Valerio Cusimano

The authors contributed equally to the research and are listed in alphabetical order

Received  February 21, 2017 Accepted  December 20, 2017 Published  March 2018

A tumor growth model accounting for angiogenic stimulation and inhibition is here considered, and a closed-loop control law is presented with the aim of tumor volume reduction by means of anti-angiogenic administration. To this end the output-feedback linearization theory is exploited, with the feedback designed on the basis of a state observer for nonlinear systems. Measurements are supposed to be acquired at discrete sampling times, and a novel theoretical development in the area of time-delay systems is applied in order to derive a continuous-time observer in spite of the presence of sampled measurements. The overall control scheme allows to set independently the control and the observer parameters thanks to the structural properties of the tumor growth model. Simulations are carried out in order to mimic a real experimental framework on mice. These results seem extremely promising: they provide very good performances according to the measurements sampling interval suggested by the experimental literature, and show a noticeable level of robustness against the observer initial estimate, as well as against the uncertainties affecting the model parameters.

Citation: Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037
##### References:
 [1] W. Arap, R. Pasqualini and E. Ruoslahti, Chemotherapy targeted to tumor vasculature, Current Opinion in Oncology, 10 (1998), 560-565.  doi: 10.1097/00001622-199811000-00014. [2] F. Cacace, A. Germani and C. Manes, State estimation and control of nonlinear systems with large and variable measurement delays, Recent Results on Nonlinear Delay Control Systems, Springer International Publishing, 4 (2016), 95–112. [3] F. Cacace, A. Germani and C. Manes, A chain observer for nonlinear systems with multiple time-varying measurement delays, SIAM Journal Control and Optimization, 52 (2014), 1862-1885.  doi: 10.1137/120876472. [4] G. Ciccarella, M. Dalla Mora and A. Germani, A Luemberger-like observer for nonlinear systems, International Journal of Control, 57 (1993), 537-556.  doi: 10.1080/00207179308934406. [5] V. Cusimano, P. Palumbo and F. Papa, Closed-loop control of tumor growth by means of antiangiogenic administration, 54th IEEE Conference on Decision and Control (CDC), Osaka, Japan, 2015,7789–7794. doi: 10.1109/CDC.2015.7403451. [6] M. Dalla Mora, A. Germani and C. Manes, Design of state observers from a drift-observability property, IEEE Transactions on Automatic Control, 45 (2000), 1536-1540.  doi: 10.1109/9.871767. [7] J. Denekamp, Vascular attack as a therapeutic strategy for cancer, Cancer and Metastasis Reviews, 9 (1990), 267-282.  doi: 10.1007/BF00046365. [8] A. D'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by hahnfeldt et al. (1999), Mathematical Biosciences, 191 (2004), 159-184.  doi: 10.1016/j.mbs.2004.06.003. [9] 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. [10] A. D'Onofrio, U. Ledzewicz, H. Maurer and H. Schattler, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222 (2009), 13-26.  doi: 10.1016/j.mbs.2009.08.004. [11] A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424.  doi: 10.1016/S0092-8240(03)00006-5. [12] J. Folkman, P. Hahnfeldt and L. Hlatky, The logic of anti-angiogenic gene therapy, The Development of Gene Therapy, Cold Sping Harbor, New York, 1998, 1–17. [13] J. Folkman, Anti-angiogenesis: New concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416.  doi: 10.1097/00000658-197203000-00014. [14] 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, 59 (1999), 4770-4775. [15] A. Isidori, Nonlinear Control Systems, Springer, London, 1995. [16] R. S. Kerbel, Inhibition of tumor angiogenesis as a strategy to circumvent acquired resistance to anti-cancer therapeutic agents, BioEssays, 13 (1991), 31-36.  doi: 10.1002/bies.950130106. [17] R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336.  doi: 10.1038/36978. [18] R. S. Kerbel, G. Klement, K. I. Pritchard and B. Kamen, Continuous low-dose anti-angiogenic/metronomic chemotherapy: From the research laboratory into the oncology clinic, Annals of Oncology, 13 (2002), 12-15.  doi: 10.1093/annonc/mdf093. [19] K. S. Kerbel and B. A. Kamen, The anti-angiogenic basis of metronomic chemotherapy, Nature Reviews Cancer, 4 (2004), 423-436.  doi: 10.1038/nrc1369. [20] M. Klagsbrun and S. Soker, VEiGF/VPF: The angiogenesis factor found?, Current Biology, 3 (1993), 699-702. [21] J. Klamka, H. Maurer and A. Swierniak, Local controllability and optimal control for a model of combined anticancer therapy with control delays, Mathematical Biosciences and Engineering, 14 (2017), 195-216. [22] L. Kovács, A. Szeles, J. Sápi, D. A. Drexler, I. Rudas, I. Harmati and Z. Sápi, Model-based angiogenic inhibition of tumor growth using modern robust control method, Computer Methods and Programs in Biomedicine, 114 (2014), e98-e110. [23] U. Ledzewicz and H. Schattler, Anti-angiogenic therapy incancer treatment as an optimal control problem, SIAM Journal on Control and Optimization, 46 (2007), 1052-1079.  doi: 10.1137/060665294. [24] U. Ledzewicz, J. Marriott, H. Maurer and H. Schattler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 157-179.  doi: 10.1093/imammb/dqp012. [25] Yu. Mikheev, V. Sobolev and E. Fridman, Asymptotic analysis of digital control systems, Automation and Remote Control, 49 (1988), 1175-1180. [26] J. Sápi, D. A. Drexler, I. Harmati, Z. Sápi and L. Kovács, Qualitative analysis of tumor growth model under antiangiogenic therapy -choosing the effective operating point and design parameters for controller design, Optimal Control Applications and Methods, 37 (2016), 848-866.  doi: 10.1002/oca.2196.

show all references

##### References:
 [1] W. Arap, R. Pasqualini and E. Ruoslahti, Chemotherapy targeted to tumor vasculature, Current Opinion in Oncology, 10 (1998), 560-565.  doi: 10.1097/00001622-199811000-00014. [2] F. Cacace, A. Germani and C. Manes, State estimation and control of nonlinear systems with large and variable measurement delays, Recent Results on Nonlinear Delay Control Systems, Springer International Publishing, 4 (2016), 95–112. [3] F. Cacace, A. Germani and C. Manes, A chain observer for nonlinear systems with multiple time-varying measurement delays, SIAM Journal Control and Optimization, 52 (2014), 1862-1885.  doi: 10.1137/120876472. [4] G. Ciccarella, M. Dalla Mora and A. Germani, A Luemberger-like observer for nonlinear systems, International Journal of Control, 57 (1993), 537-556.  doi: 10.1080/00207179308934406. [5] V. Cusimano, P. Palumbo and F. Papa, Closed-loop control of tumor growth by means of antiangiogenic administration, 54th IEEE Conference on Decision and Control (CDC), Osaka, Japan, 2015,7789–7794. doi: 10.1109/CDC.2015.7403451. [6] M. Dalla Mora, A. Germani and C. Manes, Design of state observers from a drift-observability property, IEEE Transactions on Automatic Control, 45 (2000), 1536-1540.  doi: 10.1109/9.871767. [7] J. Denekamp, Vascular attack as a therapeutic strategy for cancer, Cancer and Metastasis Reviews, 9 (1990), 267-282.  doi: 10.1007/BF00046365. [8] A. D'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by hahnfeldt et al. (1999), Mathematical Biosciences, 191 (2004), 159-184.  doi: 10.1016/j.mbs.2004.06.003. [9] 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. [10] A. D'Onofrio, U. Ledzewicz, H. Maurer and H. Schattler, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222 (2009), 13-26.  doi: 10.1016/j.mbs.2009.08.004. [11] A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424.  doi: 10.1016/S0092-8240(03)00006-5. [12] J. Folkman, P. Hahnfeldt and L. Hlatky, The logic of anti-angiogenic gene therapy, The Development of Gene Therapy, Cold Sping Harbor, New York, 1998, 1–17. [13] J. Folkman, Anti-angiogenesis: New concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416.  doi: 10.1097/00000658-197203000-00014. [14] 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, 59 (1999), 4770-4775. [15] A. Isidori, Nonlinear Control Systems, Springer, London, 1995. [16] R. S. Kerbel, Inhibition of tumor angiogenesis as a strategy to circumvent acquired resistance to anti-cancer therapeutic agents, BioEssays, 13 (1991), 31-36.  doi: 10.1002/bies.950130106. [17] R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336.  doi: 10.1038/36978. [18] R. S. Kerbel, G. Klement, K. I. Pritchard and B. Kamen, Continuous low-dose anti-angiogenic/metronomic chemotherapy: From the research laboratory into the oncology clinic, Annals of Oncology, 13 (2002), 12-15.  doi: 10.1093/annonc/mdf093. [19] K. S. Kerbel and B. A. Kamen, The anti-angiogenic basis of metronomic chemotherapy, Nature Reviews Cancer, 4 (2004), 423-436.  doi: 10.1038/nrc1369. [20] M. Klagsbrun and S. Soker, VEiGF/VPF: The angiogenesis factor found?, Current Biology, 3 (1993), 699-702. [21] J. Klamka, H. Maurer and A. Swierniak, Local controllability and optimal control for a model of combined anticancer therapy with control delays, Mathematical Biosciences and Engineering, 14 (2017), 195-216. [22] L. Kovács, A. Szeles, J. Sápi, D. A. Drexler, I. Rudas, I. Harmati and Z. Sápi, Model-based angiogenic inhibition of tumor growth using modern robust control method, Computer Methods and Programs in Biomedicine, 114 (2014), e98-e110. [23] U. Ledzewicz and H. Schattler, Anti-angiogenic therapy incancer treatment as an optimal control problem, SIAM Journal on Control and Optimization, 46 (2007), 1052-1079.  doi: 10.1137/060665294. [24] U. Ledzewicz, J. Marriott, H. Maurer and H. Schattler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 157-179.  doi: 10.1093/imammb/dqp012. [25] Yu. Mikheev, V. Sobolev and E. Fridman, Asymptotic analysis of digital control systems, Automation and Remote Control, 49 (1988), 1175-1180. [26] J. Sápi, D. A. Drexler, I. Harmati, Z. Sápi and L. Kovács, Qualitative analysis of tumor growth model under antiangiogenic therapy -choosing the effective operating point and design parameters for controller design, Optimal Control Applications and Methods, 37 (2016), 848-866.  doi: 10.1002/oca.2196.
Graphical comparison of the real and estimate state under the action of the closed loop control law
Percentage of successes for the three criteria, in a population of 1000 mice treated with endostatin
Model parameters
 $\lambda$ day$^{-1}$ $b$ day$^{-1}$ $d$ day$^{-1}$ $c$ day$^{-1}$ $\eta$ day$^{-1}$ 0.192 5.85 0.00873 0.66 1.7
 $\lambda$ day$^{-1}$ $b$ day$^{-1}$ $d$ day$^{-1}$ $c$ day$^{-1}$ $\eta$ day$^{-1}$ 0.192 5.85 0.00873 0.66 1.7
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