1. On Viable Strategy in Mathematical Model of Cancer Therapy
Alexander Bratus, Igor Samokhin
Lomonosov Moscow State University
BIOMAT 2017

2. Problem Statement
We study a dynamic optimization problem for a general nonlinear mathematical model for therapy of a lethal form of cancer.
Let us denote:
𝐶 𝑡 − the number of cancer cells
𝑁 𝑡 − the number of normal cells
𝑔 𝑡 − the nutrition concentration (oxygen, glucose, etc.)
ℎ 𝑡 −the concentration of the drug
𝑢 𝑡 − the measurable control function:
0≤𝑢 𝑡 ≤𝑀, 𝑀>0

3. Problem Statement
(1) The cell growth rates are determined by the Gompertz law: 𝑟 𝑖 𝐺 𝑔 = 𝑟 𝑖 𝑔 𝜈 𝑖 +𝑔 (2) The therapeutic agent (drug) is spread out through blood vessels 𝑑ℎ 𝑑𝑡 ~ 𝑔 𝑡 𝑢+…, 0≤𝑢 𝑡 ≤𝑀 (3) The negative influence of the drug: 𝑓 ℎ = 𝑘ℎ 𝑎+ℎ , 𝑘,𝑎>0 𝑓(ℎ) is a therapy function for cancer cells and the damage function for normal cells (4) The cancer and normal cells compete with each other for common resources −𝛽𝑁 𝑡 𝐶(𝑡) (5) Total amount of therapeutic agent is limited 0 𝑇 ℎ(𝑡) 𝑑𝑡≤𝑄, 𝑄=𝑐𝑜𝑛𝑠𝑡>0

4. Problem Statement: System of Equations
𝑑𝐶 𝑡 𝑑𝑡 = 𝑟 1 𝐺 𝑔(𝑡) 𝑙𝑛 𝑎 𝑐 − 𝑙𝑛𝐶 𝑡 𝐶 𝑡 − 𝛾 1 𝐶 𝑡 − 𝑘 1 𝑓 ℎ 𝑡 𝐶 𝑡 𝑑𝑁 𝑡 𝑑𝑡 = 𝑟 2 𝐺 𝑔(𝑡) ln 𝑎 𝑛 − 𝑙𝑛𝑁 𝑡 𝑁 𝑡 − 𝛾 2 𝑁 𝑡 − 𝑘 2 𝑓 ℎ(𝑡) 𝑁 𝑡 −𝛽𝑁 𝑡 𝐶(𝑡) 𝑑𝑔 𝑡 𝑑𝑡 =𝛼 − 𝛾 3 𝑔 𝑡 − 𝜀 1 𝐶 𝑡 + 𝜀 2 𝑁 𝑡 𝑔(𝑡) 𝑑ℎ 𝑡 𝑑𝑡 =𝑔 𝑡 𝑢(𝑡)− 𝛾 4 ℎ 𝑡 − 𝜀 3 𝐶 𝑡 + 𝜀 4 𝑁 𝑡 ℎ(𝑡)

5. Problem Statement: Therapeutic Strategy
Here, we denote: 𝑐 𝑡 =𝑙𝑛𝐶 𝑡 , 𝑛 𝑡 =𝑙𝑛𝑁 𝑡 𝑉={ 𝑐,𝑛 ∈ ℝ + 2 : 𝑐≤ 𝑐 , 𝑛≥ 𝑛 } The main goal is to find such therapeutic strategy (where control function 0≤𝑢 𝑡 ≤𝑀), that maximizes the viability time, i.e. , the time of staying in the safety region 𝑉 over all state trajectories to the system, under the condition: 0 𝑇 𝑢(𝑡) 𝑑𝑡≤𝑄, 𝑄>0

6. Steady-state Analysis
An illustration of the fact that, even if 𝑐 ∗ , 𝑛 ∗ ∈ 𝑉 , the safety region can be left for some time. Here the parameters are taken as 𝛽= 10 −6 , 𝑘 1 =7, 𝑘 2 =1.45, 𝛿 1 =0.3, 𝛿 2 =0.45, the constant control is 𝑢 = 0.1, and three different initial coordinate pairs 𝑐 0 ; 𝑛 0 = 𝐼 1 , 𝐼 2 , 𝐼 3 are considered with fixed 𝑔 0 = 1, ℎ 0 = 0.

7. Auxiliary Parameter Estimates
𝑘 1 is the therapy parameter: − 𝑘 1 𝑓 ℎ 𝐶 𝑡
𝑘 2 is damage parameter: − 𝑘 2 𝑓 ℎ 𝑁(𝑡)
To kill 75% of cancer cells and preserve 50% of normal cells it is needed:
𝑘 1 𝑘 2 =3.79
If 𝑘 1 𝑘 2 =2 then we can to fill 75% of cancer cells, but preserve only 27% of normal cells

8. Optimality Principal
Let 𝑇 be a viable time. The optimal trajectory satisfies either 𝑐 𝑇 , 𝑛 𝑇 = 𝑐 , 𝑛 Or 𝑐 𝑇 = 𝑐 , 𝑛 𝑇 > 𝑛 The second case is possible only if 0 𝑇 𝑢(𝑡) 𝑑𝑡=𝑄

9. Numerical Simulations
The results of numerical simulations are very sensitive to parameter 𝛽 which described the degree of competition between cancer and normal cells (−𝛽𝑁 𝑡 𝐶(𝑡))
Illustrations: the optimal process approximated by using the software package BOCOP for the parameter values 𝛽=10 −6 , 𝑄=0.14 (Case 1). The maximum viability time is 𝑇 1 = The integral constraint does not become active.

10. Numerical Simulations

11. Numerical Simulations
Illustrations: the optimal process approximated by using the software package BOCOP for the parameter values 𝛽=10 −6 , 𝑄=0.1 (Case 1I). The maximum viability time is 𝑇 2 = The integral constraint becomes active

12. Numerical Simulations

13. Numerical Simulations
Illustrations: the optimal process approximated by using the software package BOCOP for the parameter values 𝛽=2∗10 −6 , 𝑄=0.11 (Case IV). The maximum viability time is 𝑇 4 = The integral constraint becomes active

14. Numerical Simulations

15. Numerical Simulations
Illustrations: the optimal process approximated by using the software package BOCOP for the parameter values 𝛽=5∗10 −6 , 𝑄=0.07 (Case V). The maximum viability time is 𝑇 5 = The integral constraint does not become active

16. Numerical Simulations

17. Conclusion
The result of numerical optimization shows the good agreement with the optimality principles
Competition coefficient β plays role of the bifurcation parameter
Significant characteristic of the model is ratio 𝑘 1 / 𝑘 2 between therapy and damage coefficient
In biomedical practice the therapy and damage function (depending of drug concentration) are not precisely known

18. References
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K. Swanson, Mathematical modeling of the growth and control of tumors, Ph.D. thesis, University of Washington (1999)
N. Stepanova, Course of the immune reaction during the development of a malignant tumour, Biophysics 24 (1980)
H. Schattler, U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies: An Application of Geometric Methods, In Interdisciplinary Applied Mathematics, Vol. 42, Springer-Verlag, New York, 2015.
A. Bratus, E. Fimmel, Y. Todorov, et al., On strategies on mathematical model for leukemia therapy, Nonlinear Analysis: Real World Applications 13 (3) (2012)

19. References
N. Komarova, D. Wodarz, Targeted Cancer Treatment in Silico: Small Molecule Inhibitors and Oncolytic Viruses, Birkhauser, 2014.
A. Bratus, S. Zaichik, Smooth solutions of the Hamilton-Jacobi-Bellman equation in a mathematical model of optimal treatment of viral infections, Di. Equat. 46 (11) (2010)
A. Bratus, Y. Todorov, I. Yegorov, et al., Solution of the feedback control problem in a mathematical model of leukaemia therapy, J. Opt. Theor. Appl. 159 (2013)
A. Bratus, S. Kovalenko, E. Fimmel, On viable therapy strategy for a mathematical spatial cancer model describing the dynamics of malignant and healthy cells, Math. Biosc. Eng. 12 (1) (2015) 1-21.
A. Bratus, I. Yegorov, D. Yurchenko, Dynamic mathematical models of therapy processes against glioma and leukemia under stochastic uncertainties, Meccanica dei Materiali e delle Strutture VI (1) (2016)