American Institute of Mathematical Sciences

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October  2021, 26(10): 5281-5304. doi: 10.3934/dcdsb.2020343

Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors

Adam Glick 1, and  Antonio Mastroberardino 2,,

1. 

Department of Nuclear Engineering, University of California at Berkeley, Berkeley, CA 94270, USA
2. 

Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, USA

* Corresponding author: Antonio Mastroberardino

Received  April 2020 Revised  August 2020 Published  October 2021 Early access  November 2020

Anti-angiogenesis therapy has been an emerging cancer treatment which may be further combined with chemotherapy to enhance overall survival of cancer patients. In this paper, we investigate a system of nonlinear ordinary differential equations describing a microenvironment consisting of host cells, tumor cells, immune cells and endothelial cells while incorporating treatment combination with chemotherapy and anti-angiogenesis therapy. We perform a dynamical systems analysis demonstrating that our model is able to capture the three phases of cancer immunoediting: elimination, equilibrium, and escape. In addition, we present transcritical bifurcations for relevant parameter values that correspond to the progression from the elimination phase to the equilibrium phase. A range of medically useful tumor doubling times were simulated to determine how combined therapy affects the tumor microenvironment over the course of a 250 day treatment. This analysis found two additional bifurcation parameters that move the system of equations from the equilibrium phase to the elimination phase. We determine that the most important aspect of an effective therapy is the activation of the anti-tumor immune response.

Keywords: Mathematical oncology, tumor microenvironment, anti-angiogenic treatment.
Mathematics Subject Classification: Primary: 92C50; Secondary: 34D20.
Citation: Adam Glick, Antonio Mastroberardino. Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5281-5304. doi: 10.3934/dcdsb.2020343
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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.  Google Scholar

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[23]

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A. E. Glick and A. Mastroberardino, An optimal control approach for the treatment of solid tumors with angiogenesis inhibitors, Mathematics, 5 (2017), 49. doi: 10.3390/math5040049.  Google Scholar

[26]

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.   Google Scholar

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R. Kim, M. Emi and K. Tanabe, Cancer immunoediting from immune surveillance to immune escape, Immunology, 121 (2007), 1-14.  doi: 10.1111/j.1365-2567.2007.02587.x.  Google Scholar

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V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology, 56 (1994), 295-321.  doi: 10.1016/S0092-8240(05)80260-5.  Google Scholar

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L. Laplane, D. Duluc, N. Larmonier, T. Pradeu and A. Bikfalvi, The multiple layers of the tumor environment, Trends in Cancer, 4 (2018), 802-809.  doi: 10.1016/j.trecan.2018.10.002.  Google Scholar

[34]

U. Ledzewicz, H. Maurer and H. Schaettler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Math. Biosci. Eng., 8 (2011), 307–323. doi: 10.3934/mbe.2011.8.307.  Google Scholar

[35]

U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a two-compartment model in cancer chemotherapy, Journal of Optimization Theory and Applications, 114 (2002), 609-637.  doi: 10.1023/A:1016027113579.  Google Scholar

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Table 1. A transcritical bifurcation occurs when $ \tau _t = 4.4069303 $ days">Figure 1.  Bifurcation diagram for tumor doubling time versus tumor cell count using default parameter values given in Table 1. A transcritical bifurcation occurs when $ \tau _t = 4.4069303 $ days
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Table 1 and $ \tau _t = 10 $ days. A transcritical bifurcation occurs when $ \alpha _{ti} = 4.852032\cdot 10^{-8} $ cell$ ^{-1} $ day$ ^{-1} $">Figure 2.  Bifurcation diagram for immune cell killing rate of tumor cells versus tumor cell count using default parameter values given in Table 1 and $ \tau _t = 10 $ days. A transcritical bifurcation occurs when $ \alpha _{ti} = 4.852032\cdot 10^{-8} $ cell$ ^{-1} $ day$ ^{-1} $
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Table 1 with initial conditions given in (40) and $ \tau _t = 10 $ days. The immune response is able to effectively eliminate the tumor cells so that the system approaches the tumor-free equilibrium solution $ \mathbb{E}_4 $ given in Tables 4">Figure 3.  Numerical simulation using default parameter values given in Table 1 with initial conditions given in (40) and $ \tau _t = 10 $ days. The immune response is able to effectively eliminate the tumor cells so that the system approaches the tumor-free equilibrium solution $ \mathbb{E}_4 $ given in Tables 4
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Table 1 with $ \tau _t = 10 $ days, $ I(0) = 1.0\cdot 10^5 $ cells and all other initial conditions given in (40). The immune response is unable to eliminate the tumor so that the system approaches the equilibrium solution $ \mathbb{E}_7 $ given in Tables 4">Figure 4.  Numerical simulation using parameter values given in Table 1 with $ \tau _t = 10 $ days, $ I(0) = 1.0\cdot 10^5 $ cells and all other initial conditions given in (40). The immune response is unable to eliminate the tumor so that the system approaches the equilibrium solution $ \mathbb{E}_7 $ given in Tables 4
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Table 1 with initial conditions given in (40) and $ \tau _t = 2 $ days. The immune response is unable to eliminate the tumor so that the system approaches the equilibrium solution $ \mathbb{E}_6 $ in Table 3, which corresponds to the equilibrium phase of immunoediting">Figure 5.  Numerical simulation using parameter values given in Table 1 with initial conditions given in (40) and $ \tau _t = 2 $ days. The immune response is unable to eliminate the tumor so that the system approaches the equilibrium solution $ \mathbb{E}_6 $ in Table 3, which corresponds to the equilibrium phase of immunoediting
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Table 1 and initial conditions given by Eqn. (40), where $ \tau _t = 2 $ days and the system is simulated for 250 days, showing a 'zoomed-in' view of Fig. 5">Figure 6.  Numerical simulations using parameter values given in Table 1 and initial conditions given by Eqn. (40), where $ \tau _t = 2 $ days and the system is simulated for 250 days, showing a 'zoomed-in' view of Fig. 5
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Tables 1$ - $2">Figure 7.  Numerical simulations for the system undergoing 6 cycles of combined chemoangiogenesis therapy described in Sect. 4 using initial conditions given in Eqn. (40). We set $ \tau_{\epsilon} = 1.25 $ days, $ \tau_{\xi} = 19.6 $ days, and $ \tau _t = 2 $ days, and all other parameter values as listed in Tables 1$ - $2
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Tables 1$ - $2">Figure 8.  Numerical simulation for the system in which chemotherapy has no affect on the immune system using initial conditions given in (40). We set $ \epsilon _i = 0 $, $ \tau_{\epsilon} = 1.25 $ days, $ \tau_{\xi} = 19.6 $ days, $ \tau _t = 2 $ days, and all other parameter values as listed in Tables 1$ - $2
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Tables 1$ - $2. The tumor has a second peak early on that is not seen in Fig. 6$ - $Fig. 8, indicating that chemotherapy which harms the immune system makes the therapy more harmful to the patient than no therapy at all">Figure 9.  Numerical simulation for the system in which chemotherapy has an adverse affect on the immune system at half the effectiveness as the tumor using initial conditions given in (40). We set $ \epsilon _i $ = $ -\frac{\epsilon _t}{2} $, $ \tau_{\epsilon} = 1.25 $ days, $ \tau_{\xi} = 19.6 $ days, $ \tau _t = 2 $ days, and all other parameter values as listed in Tables 1$ - $2. The tumor has a second peak early on that is not seen in Fig. 6$ - $Fig. 8, indicating that chemotherapy which harms the immune system makes the therapy more harmful to the patient than no therapy at all
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Figure 10.  Numerical simulations of the system with therapy for (a) $ \tau _{\epsilon} = 0.417 $ days and $ \tau _{\xi} = 0.833 $ day, (b) $ \tau _{\epsilon} = 2.08 $ days and $ \tau _{\xi} = 0.833 $ days, (c) $ \tau _{\epsilon} = 0.417 $ days and $ \tau _{\xi} = 19.6 $ days, (d) $ \tau _{\epsilon} = 2.08 $ days and $ \tau _{\xi} = 19.6 $ days
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Table 1.  Values of all relevant parameters without therapy
Parameter Description Value Units Source
$ r_h $ Host cell growth parameter 1.8$ \cdot 10^{-1} $ day$ ^{-1} $ [40]
$ r_t $ Tumor growth parameter $ \frac{\ln(2)}{\tau _t} $ day$ ^{-1} $ estimated
$ \tau _t $ Tumor doubling time 2 $ - $ 10 day estimated
$ r_e $ Endothelial cell growth parameter 2.15$ \cdot 10^{-1} $ day$ ^{-1} $ [60]
$ \rho_{et} $ Tumor (VEGF) recruitment of endothelial cells 9.22$ \cdot 10^{2} $ day$ ^{-1} $ [25]
$ \sigma_i $ Influx of immune cells 1.0$ \cdot10^{4} $ cell day$ ^{-1} $ [32]
$ \delta_i $ Immune cell natural death rate 7.0$ \cdot10^{-3} $ day$ ^{-1} $ [50]
$ b_h $ Inverse of host cell carrying capacity 1.0$ \cdot10^{-9} $ cell$ ^{-1} $ [40]
$ \gamma_{te} $ Tumor carrying capacity dependence parameter 0.8 no units estimated
$ K_{te} $ Tumor cell carrying capacity 1.0$ \cdot 10^{6} $ cells [25]
$ b_e $ Inverse of endothelial cell carrying capacity 1.0$ \cdot10^{-7} $ cell$ ^{-1} $ [25]
$ \alpha_{ht} $ Host cell killing rate by tumor cells 4.8$ \cdot 10^{-10} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \alpha_{ti} $ Immune cell response to tumor cell presence 1.101$ \cdot 10^{-7} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \alpha_{it} $ Linear immune cell inactivation rate by tumor cells 2.8$ \cdot 10^{-9} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \beta_{it} $ Quadratic immune cell inactivation rate by tumor cells 3.2$ \cdot 10^{-8} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
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Parameter Description Value Units Source
$ r_h $ Host cell growth parameter 1.8$ \cdot 10^{-1} $ day$ ^{-1} $ [40]
$ r_t $ Tumor growth parameter $ \frac{\ln(2)}{\tau _t} $ day$ ^{-1} $ estimated
$ \tau _t $ Tumor doubling time 2 $ - $ 10 day estimated
$ r_e $ Endothelial cell growth parameter 2.15$ \cdot 10^{-1} $ day$ ^{-1} $ [60]
$ \rho_{et} $ Tumor (VEGF) recruitment of endothelial cells 9.22$ \cdot 10^{2} $ day$ ^{-1} $ [25]
$ \sigma_i $ Influx of immune cells 1.0$ \cdot10^{4} $ cell day$ ^{-1} $ [32]
$ \delta_i $ Immune cell natural death rate 7.0$ \cdot10^{-3} $ day$ ^{-1} $ [50]
$ b_h $ Inverse of host cell carrying capacity 1.0$ \cdot10^{-9} $ cell$ ^{-1} $ [40]
$ \gamma_{te} $ Tumor carrying capacity dependence parameter 0.8 no units estimated
$ K_{te} $ Tumor cell carrying capacity 1.0$ \cdot 10^{6} $ cells [25]
$ b_e $ Inverse of endothelial cell carrying capacity 1.0$ \cdot10^{-7} $ cell$ ^{-1} $ [25]
$ \alpha_{ht} $ Host cell killing rate by tumor cells 4.8$ \cdot 10^{-10} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \alpha_{ti} $ Immune cell response to tumor cell presence 1.101$ \cdot 10^{-7} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \alpha_{it} $ Linear immune cell inactivation rate by tumor cells 2.8$ \cdot 10^{-9} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
$ \beta_{it} $ Quadratic immune cell inactivation rate by tumor cells 3.2$ \cdot 10^{-8} $ cell$ ^{-1} $ day$ ^{-1} $ [40]
Table 2.  Values of parameters for combined therapy
Parameter Description Value Units Source
$ \xi_t $ AAT effect on tumor carrying capacity 8.9$ \cdot 10^{-1} $ no units estimated
$ \xi_e $ AAT effect on endothelial cells 9.088$ \cdot 10^{-1} $ no units estimated
$ \lambda_{\xi} $ Clearance rate of AAT agent $ \frac{ \ln(2) }{ \tau_{\xi}} $ day$ ^{-1} $ [15]
$ \tau_{\xi} $ Half life of AAT agent 0.833 $ - $ 19.6 day [61] [27]
$ \epsilon_h $ Chemotherapy effect on immune cells $ \frac{\epsilon_t}{2} $ day$ ^{-1} $ estimated
$ \epsilon_i $ Chemotherapy effect on immune cells 4.999$ \cdot 10^{-2} $ day$ ^{-1} $ estimated
$ \epsilon_t $ Chemotherapy effect on tumor cells 7.494$ \cdot 10^{-2} $ day$ ^{-1} $ estimated
$ \epsilon_e $ Chemotherapy effect on endothelial cells $ \frac{\epsilon_t}{2} $ day$ ^{-1} $ estimated
$ \lambda_{\epsilon} $ Clearance rate of chemotherapy agent $ \frac{ \ln(2) }{ \tau_{\epsilon}} $ day$ ^{-1} $ [15]
$ \tau_{\epsilon} $ Half life of chemotherapy agent 0.417 $ - $ 2.08 day [61]
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Parameter Description Value Units Source
$ \xi_t $ AAT effect on tumor carrying capacity 8.9$ \cdot 10^{-1} $ no units estimated
$ \xi_e $ AAT effect on endothelial cells 9.088$ \cdot 10^{-1} $ no units estimated
$ \lambda_{\xi} $ Clearance rate of AAT agent $ \frac{ \ln(2) }{ \tau_{\xi}} $ day$ ^{-1} $ [15]
$ \tau_{\xi} $ Half life of AAT agent 0.833 $ - $ 19.6 day [61] [27]
$ \epsilon_h $ Chemotherapy effect on immune cells $ \frac{\epsilon_t}{2} $ day$ ^{-1} $ estimated
$ \epsilon_i $ Chemotherapy effect on immune cells 4.999$ \cdot 10^{-2} $ day$ ^{-1} $ estimated
$ \epsilon_t $ Chemotherapy effect on tumor cells 7.494$ \cdot 10^{-2} $ day$ ^{-1} $ estimated
$ \epsilon_e $ Chemotherapy effect on endothelial cells $ \frac{\epsilon_t}{2} $ day$ ^{-1} $ estimated
$ \lambda_{\epsilon} $ Clearance rate of chemotherapy agent $ \frac{ \ln(2) }{ \tau_{\epsilon}} $ day$ ^{-1} $ [15]
$ \tau_{\epsilon} $ Half life of chemotherapy agent 0.417 $ - $ 2.08 day [61]
Table 3.  Relevant equilibrium solutions without therapy using default parameter values given in Table 1 and $ \tau _t = 2 $ days
Equilibrium Solution Eigenvalues
$ \mathbb{E}_1 = (0, 1.429\cdot 10^6, 0, 0) $ $ (-0.007, 0.18, 0.19, 0.215) $
$ \mathbb{E}_2 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 0) $ $ (-0.18, -0.007, 0.19, 0.215) $
$ \mathbb{E}_3 = (0, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-0.215, -0.007, 0.18, 0.19) $
$ \mathbb{E}_4 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-.215, -.18, -0.007, 0.19) $
$ \mathbb{E}_5 = (9.962\cdot 10^8, 3.126\cdot 10^6, 1.422\cdot 10^6, 2.520\cdot 10^{8}) $ $ (-10.62, -0.18, -0.0022 \pm 0.035i) $
$ \mathbb{E}_6 = (9.204\cdot 10^8, 3.045\cdot 10^6, 2.986\cdot 10^7, 1.137\cdot 10^{9}) $ $ (-48.67, -0.17, -0.16, 0.16) $
$ \mathbb{E}_7 = (0, 1.48\cdot 10^{-1}, 2.746\cdot 10^{10}, 3.432\cdot 10^{10}) $ $ (-67464.4, -1475.64, -13, -0.17) $
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Equilibrium Solution Eigenvalues
$ \mathbb{E}_1 = (0, 1.429\cdot 10^6, 0, 0) $ $ (-0.007, 0.18, 0.19, 0.215) $
$ \mathbb{E}_2 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 0) $ $ (-0.18, -0.007, 0.19, 0.215) $
$ \mathbb{E}_3 = (0, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-0.215, -0.007, 0.18, 0.19) $
$ \mathbb{E}_4 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-.215, -.18, -0.007, 0.19) $
$ \mathbb{E}_5 = (9.962\cdot 10^8, 3.126\cdot 10^6, 1.422\cdot 10^6, 2.520\cdot 10^{8}) $ $ (-10.62, -0.18, -0.0022 \pm 0.035i) $
$ \mathbb{E}_6 = (9.204\cdot 10^8, 3.045\cdot 10^6, 2.986\cdot 10^7, 1.137\cdot 10^{9}) $ $ (-48.67, -0.17, -0.16, 0.16) $
$ \mathbb{E}_7 = (0, 1.48\cdot 10^{-1}, 2.746\cdot 10^{10}, 3.432\cdot 10^{10}) $ $ (-67464.4, -1475.64, -13, -0.17) $
Table 4.  Relevant equilibrium solutions without therapy using default parameter values given in Table 1 and $ \tau _t = 10 $ days
Equilibrium Solution Eigenvalues
$ \mathbb{E}_1 = (0, 1.429\cdot 10^6, 0, 0) $ $ (-0.088, -0.007, 0.18, 0.215) $
$ \mathbb{E}_2 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 0) $ $ (-0.18, -0.088, -0.007, 0.215) $
$ \mathbb{E}_3 = (0, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-0.215, -0.088, -0.007, 0.18) $
$ \mathbb{E}_4 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-.215, -.18, -0.088, -0.007) $
$ \mathbb{E}_6 = (9.085\cdot 10^8, 6.074\cdot 10^5, 3.433\cdot 10^7, 1.218\cdot 10^{9}) $ $ (-52.17, -0.16, -0.097, 0.079) $
$ \mathbb{E}_7 = (0, 1.48\cdot 10^{-1}, 2.746\cdot 10^{10}, 3.432\cdot 10^{10}) $ $ (-67464.3, -1475.5, -13, -0.035) $
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Equilibrium Solution Eigenvalues
$ \mathbb{E}_1 = (0, 1.429\cdot 10^6, 0, 0) $ $ (-0.088, -0.007, 0.18, 0.215) $
$ \mathbb{E}_2 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 0) $ $ (-0.18, -0.088, -0.007, 0.215) $
$ \mathbb{E}_3 = (0, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-0.215, -0.088, -0.007, 0.18) $
$ \mathbb{E}_4 = (1.000\cdot 10^9, 1.429\cdot 10^6, 0, 1.000\cdot 10^7) $ $ (-.215, -.18, -0.088, -0.007) $
$ \mathbb{E}_6 = (9.085\cdot 10^8, 6.074\cdot 10^5, 3.433\cdot 10^7, 1.218\cdot 10^{9}) $ $ (-52.17, -0.16, -0.097, 0.079) $
$ \mathbb{E}_7 = (0, 1.48\cdot 10^{-1}, 2.746\cdot 10^{10}, 3.432\cdot 10^{10}) $ $ (-67464.3, -1475.5, -13, -0.035) $
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