# American Institute of Mathematical Sciences

January  2018, 23(1): 425-441. doi: 10.3934/dcdsb.2018029

## Treatment of glioma with virotherapy and TNF-α inhibitors: Analysis as a dynamical system

 1 Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland 2 Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1653, USA 3 Dept. of Electrical and Systems Engr., Washington University, St. Louis, Mo, 63130, USA

* Corresponding author

Received  August 2016 Published  January 2018

Oncolytic viruses are genetically altered replication-competent vi-ruses which upon death of a cancer cell produce many new viruses that then infect neighboring tumor cells. A mathematical model for virotherapy of glioma is analyzed as a dynamical system for the case of constant viral infusions and TNF-α inhibitors. Aside from a tumor free equilibrium point, the system also has positive equilibrium point solutions. We investigate the number of equilibrium point solutions depending on the burst number, i.e., depending on the number of new viruses that are released from a dead cancer cell and then infect neighboring tumor cells. After a transcritical bifurcation with a positive equilibrium point solution, the tumor free equilibrium point becomes asymptotically stable and if the average viral load in the system lies above a threshold value related to the transcritical bifurcation parameter, the tumor size shrinks to zero exponentially. Other bifurcation events such as saddle-node and Hopf bifurcations are explored numerically.

Citation: Elzbieta Ratajczyk, Urszula Ledzewicz, Maciej Leszczyński, Heinz Schättler. Treatment of glioma with virotherapy and TNF-α inhibitors: Analysis as a dynamical system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 425-441. doi: 10.3934/dcdsb.2018029
##### References:
 [1] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge University Press, Cambridge, 1993. [2] B. Auffinger, U. Ahmed and S. Lesniak, Oncolytic virotherapy for malignant glioma: Translating laboratory insights into clinical practice, Front. Oncol., 252 (2008), 109-122.  doi: 10.3389/fonc.2013.00032. [3] Z. Bajzer, T. Carr, K. Josić, S. J. Russell and D. Dingli, Modeling of cancer virotherapy with recombinant viruses, J. Theoretical Biology, 252 (2008), 109-122.  doi: 10.1016/j.jtbi.2008.01.016. [4] M. Biesecker, J. H. Kimn, H. Lu, D. Dingli and Z. Bajzer, Optimization of virotherapy for cancer, Bull. Math. Biology, 72 (2010), 469-489.  doi: 10.1007/s11538-009-9456-0. [5] E. A. Chiocca, Oncolytic viruses, Nat. Rev. Cancer, 2 (2002), 938-950. [6] B. S. Choudhury and B. Nasipuri, Efficient virotherapy of cancer in the presence of immune response, Int. J. Dynamics and Control, 2 (2014), 314-325.  doi: 10.1007/s40435-013-0035-8. [7] J. J. Crivelli, J. Földes, P. S. Kim and J. Wares, A mathematical model for cell-cycle specific cancer virotherapy, J. of Biological Dynamics, 6 (2012), 104-120.  doi: 10.1080/17513758.2011.613486. [8] A. El-alami Laaroussi, M. El Hia, M. Rachik, E. Benlahmar and Z. Rachik, Analysis of a mathematical model for treatment of cancer with oncolytic virotherapy, Appl. Math. Sci., 8 (2014), 929-940.  doi: 10.12988/ams.2014.311663. [9] A. Friedman, J. Tian, G. Fulci, E. Chioca and J. Wang, Glioma virotherapy: Effects of innate immune suppression and increased viral replication capacity, Can. Res., 66 (2006), 2314-2319.  doi: 10.1158/0008-5472.CAN-05-2661. [10] G. Fulci, L. Breymann, D. Gianni, K. Kurozomi, S. S. Rhee, J. Yu, B. Kaur, D. N. Louis, R. Weissleder, M. A. Caligiuri and E. A. Chiocca, Cyclophosphamide enhances glioma virotherapy by inhibiting innate immune responses, Proc. of the National Academy of Sciences -PNAS, 103 (2006), 12873-12878.  doi: 10.1073/pnas.0605496103. [11] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Verlag, New York, 1983.  doi: 10.1007/978-1-4612-1140-2. [12] N. L. Komarova and D. Wodarz, Targeted Cancer Treatment in Silico -Small Molecule inhibitors and Oncolytic Viruses, Birkhäuser, (2014).  doi: 10.1007/978-1-4614-8301-4. [13] L. R. Paiva, C. Binny, S. C. Ferreira jr and M. L. Martins, A multiscale mathematical model for oncolytic virotherapy, Can. Res., 69 (2009), 1205-1211.  doi: 10.1158/0008-5472.CAN-08-2173. [14] E. Ratajczyk, U. Ledzewicz, M. Leszczyński and A. Friedman, The role of TNF-α Inhibitor in Glioma virotherapy: A mathematical model, Math. Biosci. and Engr. -MBE, 14 (2017), 305-319.  doi: 10.3934/mbe.2017020. [15] D. Wodarz, Viruses as antitumor weapons: Defining conditions for tumor remission, Can. Res., 61 (2001), 3501-3507.

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##### References:
 [1] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge University Press, Cambridge, 1993. [2] B. Auffinger, U. Ahmed and S. Lesniak, Oncolytic virotherapy for malignant glioma: Translating laboratory insights into clinical practice, Front. Oncol., 252 (2008), 109-122.  doi: 10.3389/fonc.2013.00032. [3] Z. Bajzer, T. Carr, K. Josić, S. J. Russell and D. Dingli, Modeling of cancer virotherapy with recombinant viruses, J. Theoretical Biology, 252 (2008), 109-122.  doi: 10.1016/j.jtbi.2008.01.016. [4] M. Biesecker, J. H. Kimn, H. Lu, D. Dingli and Z. Bajzer, Optimization of virotherapy for cancer, Bull. Math. Biology, 72 (2010), 469-489.  doi: 10.1007/s11538-009-9456-0. [5] E. A. Chiocca, Oncolytic viruses, Nat. Rev. Cancer, 2 (2002), 938-950. [6] B. S. Choudhury and B. Nasipuri, Efficient virotherapy of cancer in the presence of immune response, Int. J. Dynamics and Control, 2 (2014), 314-325.  doi: 10.1007/s40435-013-0035-8. [7] J. J. Crivelli, J. Földes, P. S. Kim and J. Wares, A mathematical model for cell-cycle specific cancer virotherapy, J. of Biological Dynamics, 6 (2012), 104-120.  doi: 10.1080/17513758.2011.613486. [8] A. El-alami Laaroussi, M. El Hia, M. Rachik, E. Benlahmar and Z. Rachik, Analysis of a mathematical model for treatment of cancer with oncolytic virotherapy, Appl. Math. Sci., 8 (2014), 929-940.  doi: 10.12988/ams.2014.311663. [9] A. Friedman, J. Tian, G. Fulci, E. Chioca and J. Wang, Glioma virotherapy: Effects of innate immune suppression and increased viral replication capacity, Can. Res., 66 (2006), 2314-2319.  doi: 10.1158/0008-5472.CAN-05-2661. [10] G. Fulci, L. Breymann, D. Gianni, K. Kurozomi, S. S. Rhee, J. Yu, B. Kaur, D. N. Louis, R. Weissleder, M. A. Caligiuri and E. A. Chiocca, Cyclophosphamide enhances glioma virotherapy by inhibiting innate immune responses, Proc. of the National Academy of Sciences -PNAS, 103 (2006), 12873-12878.  doi: 10.1073/pnas.0605496103. [11] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Verlag, New York, 1983.  doi: 10.1007/978-1-4612-1140-2. [12] N. L. Komarova and D. Wodarz, Targeted Cancer Treatment in Silico -Small Molecule inhibitors and Oncolytic Viruses, Birkhäuser, (2014).  doi: 10.1007/978-1-4614-8301-4. [13] L. R. Paiva, C. Binny, S. C. Ferreira jr and M. L. Martins, A multiscale mathematical model for oncolytic virotherapy, Can. Res., 69 (2009), 1205-1211.  doi: 10.1158/0008-5472.CAN-08-2173. [14] E. Ratajczyk, U. Ledzewicz, M. Leszczyński and A. Friedman, The role of TNF-α Inhibitor in Glioma virotherapy: A mathematical model, Math. Biosci. and Engr. -MBE, 14 (2017), 305-319.  doi: 10.3934/mbe.2017020. [15] D. Wodarz, Viruses as antitumor weapons: Defining conditions for tumor remission, Can. Res., 61 (2001), 3501-3507.
Illustration of Proposition 3 for $c=0$
'Graph' of polynomial $P$ if $w^0 < 1$ (top) and in the two subcases that arise for $w^0>1$ (bottom)
Illustration of Propositions 3 and 4
Transcritical bifurcation
Values of the tumor free and positive equilibrium points for $B=90$ (left), $B=125$ (middle) and $B=140$ (right) plotted as function of $C$. In each case there is a saddle-node bifurcation which generates an upper ($z_u$) and lower branch $(z_\ell)$ of positive solutions. The upper branch only exists for the short range and terminates as $\bar{y}=0.1=\frac{\delta_M}{s}$ which causes $\bar{M}\rightarrow \infty$. The lower branch exists until $C=2.5$ and terminates as $\bar{y}$ becomes zero
Periodic orbit for $C=0.3$ and $B=140$
States and parameters of the model
 States Description Dimension Num. Value(s) x density of uninfected cancer stem cells $\frac{g}{{c{m^3}}}$ y density of infected cancer stem cells $\frac{g}{{c{m^3}}}$ M density of macrophages $\frac{g}{{c{m^3}}}$ T density of TNF-α $\frac{g}{{c{m^3}}}$ v density of virus $\frac{g}{{c{m^3}}}$ z z = (x; y; M; T; v#8224; Parameter Description Dimension Num. Value(s) α proliferation rate of uninfected tumor cells 1/day 0.2 β infection rate of tumor cells by viruses $\frac{{c{m^3}}}{{g \cdot day}}$ 2·104 ρ rate of loss of viruses during infection $ρ$ 0.04 k effectiveness of the inhibitory action of TNF-α 1/day 0.4 δy infected tumor cell death rate 1/day 0.2 λ TNF-α production rate 1/day 2.86·10-3 δT TNF-α cell degradation rate 1/day 55.45 δM macrophages death rate 1/day 0.015 b burst size of infected cells during apoptosis ×10-6 50 - 150 KT carrying capacity of TNF-α $\frac{g}{{c{m^3}}}$ 5·10-7 κ degradation of TNF-α due to its action on infected cells 1/day 4·10-10 δv virus lysis rate 1/day 0.5 A constant source of macrophages $\frac{g}{{c{m^3} \cdot day}}$ 0.9·10-6 s stimulation rate of macrophages by infected cells $\frac{{c{m^3}}}{{g \cdot day}}$ 0.15 δx death rate of uninfected cancer cells 1/day 0.1 c constant infusion of the virus $\frac{{g \times {{10}^{-6}}}}{{c{m^3} \cdot day}}$ 0 - 150 d constant infusion of the TNF-α inhibitor
 States Description Dimension Num. Value(s) x density of uninfected cancer stem cells $\frac{g}{{c{m^3}}}$ y density of infected cancer stem cells $\frac{g}{{c{m^3}}}$ M density of macrophages $\frac{g}{{c{m^3}}}$ T density of TNF-α $\frac{g}{{c{m^3}}}$ v density of virus $\frac{g}{{c{m^3}}}$ z z = (x; y; M; T; v#8224; Parameter Description Dimension Num. Value(s) α proliferation rate of uninfected tumor cells 1/day 0.2 β infection rate of tumor cells by viruses $\frac{{c{m^3}}}{{g \cdot day}}$ 2·104 ρ rate of loss of viruses during infection $ρ$ 0.04 k effectiveness of the inhibitory action of TNF-α 1/day 0.4 δy infected tumor cell death rate 1/day 0.2 λ TNF-α production rate 1/day 2.86·10-3 δT TNF-α cell degradation rate 1/day 55.45 δM macrophages death rate 1/day 0.015 b burst size of infected cells during apoptosis ×10-6 50 - 150 KT carrying capacity of TNF-α $\frac{g}{{c{m^3}}}$ 5·10-7 κ degradation of TNF-α due to its action on infected cells 1/day 4·10-10 δv virus lysis rate 1/day 0.5 A constant source of macrophages $\frac{g}{{c{m^3} \cdot day}}$ 0.9·10-6 s stimulation rate of macrophages by infected cells $\frac{{c{m^3}}}{{g \cdot day}}$ 0.15 δx death rate of uninfected cancer cells 1/day 0.1 c constant infusion of the virus $\frac{{g \times {{10}^{-6}}}}{{c{m^3} \cdot day}}$ 0 - 150 d constant infusion of the TNF-α inhibitor
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